This study introduces a novel Topological Derivative-based Sensitivity Analysis (TDSA) methodology for three-dimensional (3D) discrete variable topology optimization. Recently, the authors pointed out that the discrete variable sensitivity can be related to the topological derivative, and thus can be rationally approximated by the specially customized topological derivative for plane stress problems. However, the 3D discrete variable sensitivity requires the 3D topological derivative with element-shaped (e.g., unit cube) domain perturbation, whose analytical solution is not available in the literature. This paper proposes a unified parameter fitting framework to calculate the 3D topological derivative with arbitrary shape 3D domain perturbation. The formula for spherical hole perturbation obtained through this parameter fitting framework is identical to the well-known analytical topological derivative formula. Further, the 3D discrete variable sensitivity can be easily obtained through element-shaped domain perturbation. With the recently proposed Sequential Approximate Integer Programming (SAIP), numerical examples demonstrate that TDSA is precise not only for the self-adjoint 3D minimum compliance problems but also for the non-adjoint 3D compliant mechanism design problems. In sum, numerical examples have been conducted by feeding the derived sensitivity to the SAIP optimization algorithms, and the correctness of the sensitivity information has been well demonstrated. This research further solidifies the foundation of rational discrete variable sensitivity analysis in 3D topology optimization.

中文翻译：

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基于拓扑导数的三维离散变量拓扑优化灵敏度分析


本研究引入了一种新颖的基于拓扑导数的灵敏度分析 (TDSA) 方法，用于三维 (3D) 离散变量拓扑优化。最近，作者指出，离散变量的灵敏度可以与拓扑导数相关，因此可以通过专门定制的平面应力问题的拓扑导数来合理地逼近。然而，3D离散变量灵敏度需要具有单元形状（例如，单位立方体）域扰动的3D拓扑导数，其解析解在文献中不可用。本文提出了一个统一的参数拟合框架来计算任意形状3D域扰动的3D拓扑导数。通过该参数拟合框架获得的球孔扰动公式与众所周知的解析拓扑导数公式相同。此外，通过元形域扰动可以轻松获得3D离散变量灵敏度。通过最近提出的顺序近似整数规划（SAIP），数值示例表明 TDSA 不仅对于自伴 3D 最小柔顺问题而且对于非伴随 3D 柔顺机构设计问题也是精确的。综上所述，通过将推导的灵敏度输入SAIP优化算法进行了数值算例，并且很好地证明了灵敏度信息的正确性。该研究进一步巩固了3D拓扑优化中理性离散变量敏感性分析的基础。