Topology- and Geometry-Exact Coupling for Incompressible Fluids and Thin Deformables

We introduce a topology-preserving discretization for coupling incompressible fluids with thin deformable structures, achieving guaranteed leakproofness through preservation of fluid domain connectivity. Our approach leverages a stitching algorithm applied to a clipped Voronoi diagram generated from Lagrangian fluid particles, in order to maintain path connectivity around obstacles. This geometric discretization naturally conforms to arbitrarily thin structures, enabling boundary conditions to be enforced exactly at fluid-solid interfaces. By discretizing the pressure projection equations on this conforming mesh, we can enforce velocity boundary conditions at the interface for the fluid while applying pressure forces directly on the solid boundary, enabling sharp two-way coupling between phases. The resulting method prevents fluid leakage through solids while permitting flow wherever a continuous path exists through the fluid domain. We demonstrate the effectiveness of our approach on diverse scenarios including flows around thin membranes, complex geometries with narrow passages, and deformable structures immersed in liquid, showcasing robust two-way coupling without artificial sealing or leakage artifacts.