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

Identifying stiffness extrema in large-scale structures is computationally expensive due to the global eigenvalue tracking required by traditional methods. This paper introduces a localized decomposition approach that enables rapid extremum assessment without full-structure analysis. The key innovation is a Domain Decomposition Theorem proving that global stiffness extrema can be predicted from localized sensitivity fields computed on structural subdomains. We establish rigorous error bounds showing that prediction accuracy depends on subdomain size and interface coupling strength, providing theoretical guidance for optimal decomposition strategies. For structures with N degrees of freedom, the method achieves O(kn³) computational complexity where k is the number of subdomains and n ≪ N is the subdomain size, resulting in order-of-magnitude speedup. We validate the approach through five case studies—including a planar truss, portal frame, arch bridge, cable network, and a 15,000-DOF cable-stayed bridge—demonstrating 13–40× computational speedup (average 22.6×) with prediction errors below 3% (average 1.8%) and computed error bounds holding in every case. The framework is particularly advantageous for preliminary design exploration and parametric studies where multiple design variants must be rapidly evaluated.