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

TL;DR

This work tackles the challenge of robustly coupling incompressible fluids with thin or codimensional solids without leaking flow through solid barriers. It introduces a topology- and geometry-exact discretization based on a clipped Voronoi diagram stitched to preserve fluid connectivity around solids, enabling sharp boundary enforcement and accurate two-way coupling. The method discretizes the pressure projection on a conforming, unstructured Voronoi mesh, enforces velocity BCs at fluid-solid interfaces, and transfers pressure forces to solids, maintaining exact flux balance and gauge consistency across topology changes. Demonstrations across membranes, narrow channels, and deformable structures show leakproofness and topology preservation across resolutions, with clear advantages over volumetric or naive clipped-Voronoi approaches in preserving valid flow paths and preventing artificial sealing.

Abstract

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.

Figures (13)

• Figure 1: Failure case of volumetrically resolving codimensional solids (b), compared to our approach (a). A constant rightward velocity field is passed through two codimensional sheets. The Voronoi diagram (coloured cells) and streamlines (red lines) are shown. Notice that when volumetrically resolving the solid, the velocity field incorrectly bends around the solid as the expanded boundary condition now must be respected. The small channel inside is also completely blocked off. Our method preserves the small channel and properly resolves the velocity field.
• Figure 2: Fluid pushing through a stack of codimensional thin geometry. Our method successfully resolves even a challenging stack of many small orifices without clogging any.
• Figure 3: Existing Voronoi meshing methods fail to mesh an immersed codimensional sheet. (a) Clipped Voronoi methods use the codimensional sheet as a clipping plane, effectively deleting parts of the cell opposite of a sheet as if it were a halfplane. (b) Constrained Voronoi methods treat vertices of the occluder as Voronoi sites, creating extra sites that are not allocated to any fluid particle. Notably, the example shown separates the domain into two separate fluid regions, breaking the original topology of the fluid domain.
• Figure 4: Stitching orphaned cells back to valid cells from the clipped Voronoi based on path-connectivity constraints.
• Figure 5: Discrete resolution-independent flux cancellation. In this configuration, the entire maze interior is represented by only two Voronoi cells (particles). Despite this coarse discretisation, summing the flux-form divergence over the path-connected maze region causes all internal face fluxes to cancel exactly, leaving only boundary fluxes at the two openings. After pressure projection, these boundary fluxes must balance, so the total flux entering the maze equals the total flux exiting it, ensuring exact flux balance between fluid regions connected only through the narrow maze.