  In the case of media comprised of impermeable particles, fluid flows through
voids around impenetrable grains. For sufficiently low concentrations of the
latter, spaces around grains join to allow transport on macroscopic scales,
whereas greater impenetrable inclusion densities disrupt void networks and
block macroscopic fluid flow. A critical grain concentration $\rho_{c}$ marks
the percolation transition or phase boundary separating these two regimes. With
a dynamical infiltration technique in which virtual tracer particles explore
void spaces, we calculate critical grain concentrations for randomly placed
interpenetrating impermeable toroidal inclusions; the latter consist of
surfaces of revolution with circular and square cross sections. In this manner,
we study for the first time continuum percolation transitions involving
non-convex grains. As the radius of revolution increases relative to the length
scale of the torus cross section, the tori develop a central hole, a
topological transition accompanied by a cusp in the critical porosity for
percolation. With a further increase in the radius of revolution, as
constituent grains become more ring-like in appearance, we find that the
critical porosity converges to that of high aspect ratio cylindrical
counterparts only for randomly oriented grains.
