  The Immersed Boundary (IB) method is a mathematical framework for
constructing robust numerical methods to study fluid-structure interaction in
problems involving an elastic structure immersed in a viscous fluid. The IB
formulation uses an Eulerian representation of the fluid and a Lagrangian
representation of the structure. The Lagrangian and Eulerian frames are coupled
by integral transforms with delta function kernels. The discretized IB
equations use approximations to these transforms with regularized delta
function kernels to interpolate the fluid velocity to the structure, and to
spread structural forces to the fluid. It is well-known that the conventional
IB method can suffer from poor volume conservation since the interpolated
Lagrangian velocity field is not generally divergence-free, and so this can
cause spurious volume changes. In practice, the lack of volume conservation is
especially pronounced for cases where there are large pressure differences
across thin structural boundaries. The aim of this paper is to greatly reduce
the volume error of the IB method by introducing velocity-interpolation and
force-spreading schemes with the properties that the interpolated velocity
field in which the structure moves is at least C1 and satisfies a continuous
divergence-free condition, and that the force-spreading operator is the adjoint
of the velocity-interpolation operator. We confirm through numerical
experiments in two and three spatial dimensions that this new IB method is able
to achieve substantial improvement in volume conservation compared to other
existing IB methods, at the expense of a modest increase in the computational
cost. Further, the new method provides smoother Lagrangian forces (tractions)
than traditional IB methods. The method presented here is restricted to
periodic computational domains. Its generalization to non-periodic domains is
important future work.
