Abstract: Fluid-solid systems are common in scientific and bio-engineering applications. The immersed boundary (IB) method is a general approach to modeling such systems. A difficulty of this IB formulation is that the pressure and viscous stress are generally discontinuous at fluid-solid interfaces. We will describe extensions of the IB method that sharply resolve stress discontinuities at fluid-solid interfaces. These methods can be viewed as extensions of the immersed interface method to non-interfacial (codimension-0) solid bodies. As an application of the technique, we present some preliminary results on bileaflet mechanical heart valve modeling.