  Topology optimization is able to maximally leverage the high DOFs and
mechanical potentiality of porous foams but faces three fundamental challenges:
conforming to free-form outer shapes, maintaining geometric connectivity
between adjacent cells, and achieving high simulation accuracy. To resolve the
issues, borrowing the concept from Voronoi tessellation, we propose to use the
site (or seed) positions and radii of the beams as the DOFs for open-cell foam
design. Such DOFs cover extensive design space and have clear geometrical
meaning, which makes it easy to provide explicit controls (e.g. granularity).
During the gradient-based optimization, the foam topology can change freely,
and some seeds may even be pushed out of the shape, which greatly alleviates
the challenges of prescribing a fixed underlying grid. The mechanical property
of our foam is computed from its highly heterogeneous density field counterpart
discretized on a background mesh, with a much improved accuracy via a new
material-aware numerical coarsening method. We also explore the
differentiability of the open-cell Voronoi foams w.r.t. its seed locations, and
propose a local finite difference method to estimate the derivatives
efficiently. We do not only show the improved foam performance of our Voronoi
foam in comparison with classical topology optimization approaches, but also
demonstrate its advantages in various settings, especially when the target
volume fraction is extremely low.
