敏感热点惩罚：一种针对一阶最坏情况扰动的鲁棒拓扑优化框架

中文摘要 AI 辅助

确定性拓扑优化可高效生成高性能结构设计，但未明确控制由制造偏差和几何不确定性引发的局部脆弱性，这类脆弱性常表现为应力集中或铰链式变形机制。传统鲁棒拓扑优化可抑制这些特征，但通常需要多个设计实现，且计算成本大幅增加。Sigmund等人（2026）近期的研究提出了敏感热点惩罚（SHoSP），其通过目标函数灵敏度的光滑最大值增强名义目标，以较低的额外成本抑制局部脆弱特征，同时促进更均匀的应力分布。本文利用泰勒展开和Hölder对偶性，建立了SHoSP与材料-质量扰动预算下一阶最坏情况鲁棒近似的通用联系，确定原始惩罚权重为相对于设计域面积（2D）或体积（3D）的无量纲范数有界扰动预算。该解释通过限制预算扰动下的最大一阶退化，说明了其对敏感热点、铰链定位和应力集中的抑制作用。本文针对柔顺性最小化和柔顺机构设计开展研究，通过松弛最坏情况密度扰动，以及在等效扰动预算下比较确定性设计与SHoSP设计来评估鲁棒性，对结构化网格观察到的应力相关效应，通过面向应力的拓扑优化和贴合体有限元分析进行交叉验证。向多孔填充优化、多载荷工况及大规模三维算例的扩展，进一步证明了SHoSP框架的适用性和卓越的可扩展性。

英文摘要

Deterministic topology optimization can efficiently generate high-performance structural designs, but it does not explicitly control localized fragility induced by manufacturing variations and geometric uncertainties. Such fragility often appears as stress concentrations or hinge-like deformation mechanisms. Conventional robust topology optimization can suppress these features, but typically requires multiple design realizations and substantially increased computational cost. Recent work by Sigmund et al. (2026) introduced sensitivity hot spot penalization (SHoSP), which augments the nominal objective by a smooth maximum of its sensitivities and suppresses localized fragile features at low additional cost while promoting more even stress distributions. This paper establishes a general connection between SHoSP and a first-order worst-case robust approximation under a material-mass perturbation budget using a Taylor expansion and H{"o}lder's duality. The original penalty weight is identified as a dimensionless norm-bounded perturbation budget relative to the area (2D) or volume (3D) of the design domain. This interpretation explains the suppression of sensitivity hot spots, hinge localization, and stress concentrations by limiting the maximum first-order degradation under budgeted perturbations. It is investigated for compliance minimization and compliant mechanism design. Robustness is assessed using a relaxed worst-case density perturbation and by comparing deterministic and SHoSP designs under equivalent perturbation budgets. Stress-related effects observed on structured meshes are cross-validated by stress-oriented topology optimization and body-fitted finite element analyses. Extensions to porous infill optimization, multiple load cases, and large-scale 3D examples further demonstrate the applicability and superior scalability of the SHoSP framework.

发表机构

• Technical University of Denmark（丹麦技术大学）

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