Figure 2:

(a) The immersed interface domain and (b) the partitioned fluid (${\Omega }_t^{\mathrm{f}}$) and solid (${\Omega }_t^{\mathrm{s}}$) subdomains. (c) In the penalty-ILE method, the configuration of the explicit representation of the fluid-structure interface (determined by χ) conforms to the boundary of the structure (determined by ξ) in an approximate sense. Specifically, the motion of the surface representation is determined by the local fluid velocity at the fluid-structure interface ${\Gamma }_t^{\mathrm{fs}}$ whereas the no-slip condition is satisfied in an approximate sense by spring-like forces that penalize displacements between the two representations of the fluid-structure interface. The displacement between the two representations is exaggerated here for illustration purposes. In practice, we always impose an assertion specifying that the maximum relative displacement is at least less than 0.1 of the Cartesian grid spacing.