  Immersed boundary methods are extensively used for simulations of dynamic
solid objects interacting with fluids due to their computational efficiency and
modelling flexibility compared to body-fitted grid methods. However, thin
geometries, such as shells and membranes, cause a violation of the boundary
conditions across the surface for many immersed boundary projection algorithms.
Using a one-dimensional analytical derivation and multi-dimensional numerical
simulations, this manuscript shows that adjustment of the Poisson matrix itself
is require to avoid large velocity, pressure, and force prediction errors when
the pressure jump across the interface is substantial and that these errors
increase with Reynolds number. A new minimal thickness modification is
developed for the Boundary Data Immersion Method (BDIM-{\sigma}),which avoids
these issues while still enabling the use of efficient projection algorithms
for high-speed immersed surface simulations.
