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.

翻译：拓扑优化能够最大化利用多孔泡沫的高自由度与力学潜力，但面临三个基本挑战：与自由形态外轮廓的共形、相邻胞元间的几何连通性保持，以及高仿真精度的实现。为解决这些问题，借鉴沃罗诺伊镶嵌的概念，我们提出将梁的站点（或种子）位置和半径作为开孔泡沫设计的自由度。此类自由度覆盖广泛设计空间且具有清晰几何含义，便于提供显式控制（如粒度）。在基于梯度的优化过程中，泡沫拓扑可自由变化，部分种子甚至可能被推出形状边界，这极大缓解了预设固定网格的挑战。泡沫的力学性能通过其离散在背景网格上的高度异质性密度场来计算，并借助一种新的材料感知数值粗化方法显著提高了计算精度。我们还探索了开孔沃罗诺伊泡沫相对其种子位置的可微性，并提出一种局部有限差分方法以高效估计导数。我们不仅展示了所提出的沃罗诺伊泡沫相较于经典拓扑优化方法的性能提升，还证明了其在多种场景下的优势，尤其是在目标体积分数极低的情况下。