Hollow structures can meet the demand of thickness control in manufacture, and achieve lightweight structures in design. However, topologically optimized results are generally solid structures, which are not easily transformed into hollow structures. Therefore, this paper proposes a three-dimensional moving morphable bar method to obtain hollow structures directly. The geometrical features of the three-dimensional solid moving morphable bars are projected onto a fixed grid, so the element density function can be obtained by a smooth Heaviside approximation of distance functions. Then the three-dimensional hollow moving morphable bars are further obtained by performing the Boolean subtraction of two solid bars. Also, the differentiable optimization model is established and the corresponding sensitivities are analytically derived. Numerical examples demonstrate that the proposed method can effectively achieve optimized designs of hollow structures. Besides, the Matlab code is provided to illustrate this method and facilitate further expansion.

中空结构可以满足制造中厚度控制的需求，并可以在设计中实现轻质结构。但是，拓扑优化的结果通常是实体结构，不容易转换为空心结构。因此，本文提出了一种三维移动变形棒法，以直接获得空心结构。三维实体移动形变棒的几何特征被投影到固定的网格上，因此可以通过距离函数的光滑Heaviside近似来获得元素密度函数。然后，通过对两个实心钢筋进行布尔减法，进一步获得三维空心可变形钢筋。此外，建立了可微分优化模型，并通过分析得出了相应的灵敏度。数值算例表明，该方法可以有效地实现空心结构的优化设计。此外，提供了Matlab代码来说明此方法并有助于进一步扩展。