Paper 2Rapid Assessment of Stiffness Extrema via Localized Decomposition: Theory and Error Bounds

Identifying stiffness extrema in large-scale structures typically requires computationally prohibitive global eigenvalue tracking. This paper introduces a "Localized Decomposition Theorem" that enables the prediction of global stiffness behavior through subdomain-level sensitivity analysis. We derive rigorous error bounds that scale with subdomain interface coupling strength, proving that global extrema can be captured with high precision (average error < 2%) while reducing computational complexity from O (N³) to O (kn³). Validation across multiple scales—from planar trusses to a 15, 000-DOF cable-stayed bridge—demonstrates a 13-40x speedup, making Mang’s criteria viable for real-time design optimization and parametric exploration. 在大规模结构中识别刚度极值通常需要极高计算成本的全结构特征值追踪。本文提出了一种“局部切分定理”, 通过子域层级的灵敏度分析实现对全局刚度行为的预测。我们推导了随子域界面耦合强度变化的严谨误差界限, 证明了在将计算复杂度从 O (N³) 降低至 O (kn³) 的同时, 仍能以高精度 (平均误差 < 2%) 捕获全局极值。通过从平面桁架到 1. 5 万自由度斜拉桥的多尺度验证, 结果显示计算速度提升了 13-40 倍, 使 Mang 准则在实时设计优化和参数化研究中具备了实用性。