Explicit Topology Optimization of Voronoi Foams

Topology optimization can maximally leverage the high DOFs and mechanical potentiality of porous foams but faces challenges in adapting to free-form outer shapes, maintaining full connectivity between adjacent foam cells, and achieving high simulation accuracy. Utilizing the concept of Voronoi tessellation may help overcome the challenges owing to its distinguished properties on highly flexible topology, natural edge connectivity, and easy shape conforming. However, a variational optimization of the so-called Voronoi foams has not yet been fully explored. In addressing the issue, a concept of explicit topology optimization of open-cell Voronoi foams is proposed that can efficiently and reliably guide the foam's topology and geometry variations under critical physical and geometric requirements. Taking the site (or seed) positions and beam radii as the DOFs, we explore the differentiability of the open-cell Voronoi foams w.r.t. its seed locations, and propose a highly efficient local finite difference method to estimate the derivatives. During the gradient-based optimization, the foam topology can change freely, and some seeds may even be pushed out of shape, which greatly alleviates the challenges of prescribing a fixed underlying grid. The foam's mechanical property is also computed with a much-improved efficiency by an order of magnitude, in comparison with benchmark FEM, via a new material-aware numerical coarsening method on its highly heterogeneous density field counterpart. We show the improved performance of our Voronoi foam in comparison with classical topology optimization approaches and demonstrate its advantages in various settings.