  Cellular structures manifest their outstanding mechanical properties in many
biological systems. One key challenge for designing and optimizing these
geometrically complicated structures lies in devising an effective geometric
representation to characterize the system's spatially varying cellular
evolution driven by objective sensitivities. A conventional discrete cellular
structure, e.g., a Voronoi diagram, whose representation relies on discrete
Voronoi cells and faces, lacks its differentiability to facilitate large-scale,
gradient-based topology optimizations. We propose a topology optimization
algorithm based on a differentiable and generalized Voronoi representation that
can evolve the cellular structure as a continuous field. The central piece of
our method is a hybrid particle-grid representation to encode the previously
discrete Voronoi diagram into a continuous density field defined in a Euclidean
space. Based on this differentiable representation, we further extend it to
tackle anisotropic cells, free boundaries, and functionally-graded cellular
structures. Our differentiable Voronoi diagram enables the integration of an
effective cellular representation into the state-of-the-art topology
optimization pipelines, which defines a novel design space for cellular
structures to explore design options effectively that were impractical for
previous approaches. We showcase the efficacy of our approach by optimizing
cellular structures with up to thousands of anisotropic cells, including femur
bone and Odonata wing.
