  Strongly coupled immersed boundary (IB) methods solve the nonlinear fluid and
structural equations of motion simultaneously for strongly enforcing the
no-slip constraint on the body. Handling this constraint requires solving
several large dimensional systems that scale by the number of grid points in
the flow domain even though the nonlinear constraints scale only by the small
number of points used to represent the fluid-structure interface. These costly
large scale operations for determining only a small number of unknowns at the
interface creates a bottleneck to efficiently time-advancing strongly coupled
IB methods. In this manuscript, we present a remedy for this bottleneck that is
motivated by the efficient strategy employed in stationary-body IB methods
while preserving the favorable stability properties of strongly coupled
algorithms -- we precompute a matrix that encapsulates the large dimensional
system so that the prohibitive large scale operations need not be performed at
every time step. This precomputation process yields a modified system of
small-dimensional constraint equations that is solved at minimal computational
cost while time advancing the equations. We also present a parallel
implementation that scales favorably across multiple processors. The accuracy,
computational efficiency and scalability of our approach are demonstrated on
several two dimensional flow problems. Although the demonstration problems
consist of a combination of rigid and torsionally mounted bodies, the
formulation is derived in a more general setting involving an arbitrary number
of rigid, torsionally mounted, and continuously deformable bodies.
