A Deterministic Geometrically Precise Algorithm for calculation of percolation thresholds through voids around impermeable faceted solids

We examine percolation phenomena in strongly disordered porous media comprised of randomly placed faceted grains in which fluid or charge flows through void volumes around the impenetrable polyhedral inclusions. Although we consider grains in the form of convex solids (e.g. the Platonic solids in aligned and randomly oriented cases), statistics related to the labyrinthine spaces between grains, whch are not in general convex, have been difficult to determine even with knowledge of the locations and orientations of constituent impermeable grains. Nevertheless, we discuss and present results for a method which does account for void networks in a geometrically precise way while at the same time allowing for consideration of large enough system sizes to compete in accuracy with existing techniques. The algorithm for mapping out the shapes of spaces between grains uses an efficient method to find vertices lining the intricate void networks which scales linearly in the simulation volume. Vertices found in this way are connect to neighboring vertices which also lie on the void volume, and these vertex networks are then used to reconstruct the faces of void volumes; care nust be taken in correctly identifiying the subset of faces which have a topological genus greater than zero, and we discuss how to navigate this challenge efficiently and correctly.