考虑屈曲约束的拓扑优化：自适应特征值聚合与模态识别
Topology optimization with buckling constraints: Adaptive eigenvalue aggregation and modality identification
- Indian Institute of Technology Bhubaneswar（印度理工学院布巴内斯瓦尔分校）
- Indian Institute of Science（印度科学学院）
- Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对带屈曲约束的拓扑优化，提出动态确定特征值聚合数量及基于特征值差的新公式以识别最优模态，显著降低计算成本并准确预测屈曲模态。
AI中文摘要：
本文提出了新颖的方法来应对拓扑优化中带屈曲约束的两个关键挑战：确定需要聚合多少个特征值，以及识别最优设计处的特征值模态。我们证明了当选择任意固定数量的特征值进行聚合时会出现数学上的不一致性。具体来说，如果临界特征值的重数小于所选的子集大小，则可能发生与高阶特征值的合并。在聚合过程中忽略这些高阶特征值会导致灵敏度计算不正确，而仅仅增加预选特征值的数量则会不必要地提高计算成本。为解决此问题，我们提出了一种方法，该方法在每次优化迭代中动态确定所需的精确特征值数量。在展示的数值实验中，所提出的方法相对于传统方法显著减少了计算时间，同时保持了有竞争力的性能。此外，准确预测最优点处的模态对于理解优化设计的屈曲模态形状至关重要。我们提出了一种新的优化公式，利用特征值之间的差异来捕捉真实的特征值模态。该公式首先在经典的1D固支柱上进行验证，成功预测了双模态解和最优特征向量。该方法表明，在给定参数集下，墙体加固问题和受剪铰支板问题的最优解分别为三模态和四模态。
英文摘要：
This paper presents novel approaches to tackle two key challenges in topology optimization problems with buckling constraints: determining how many eigenvalues to aggregate and identifying the eigenvalue modality at the optimal design. We demonstrate the mathematical inconsistencies that arise when choosing an arbitrary fixed number of eigenvalues for aggregation. Specifically, if the multiplicity of the critical eigenvalues is smaller than the chosen subset size, coalescence with higher-order eigenvalues can occur. Neglecting these during aggregation causes incorrect sensitivities, while simply increasing the preselected eigenvalue count unnecessarily raises computational costs. To resolve this issue, we propose an approach that dynamically determines the exact number of eigenvalues required at each optimization iteration. The proposed approach reduces computation time significantly while maintaining competitive performance relative to the conventional method across the numerical experiments presented. Additionally, accurately predicting modality at the optimal point is crucial for understanding the buckled mode shapes of optimized designs. A new optimization formulation is proposed using the difference between eigenvalues to capture the true eigenvalue modality. This formulation is first validated on a classical 1D clamped column, successfully predicting both the bimodal solution and the optimal eigenvectors. The approach demonstrates that the optimal solutions for the wall reinforcement and shear-loaded hinged plate problems for the given set of parameters are trimodal and tetramodal, respectively.
