Investigating the Numerical Stability of Dynamic Relaxation Methods with Automatic Load-Increment Schemes to Improve Snap-Back Prediction

Automatic load incrementation schemes tailor-made for the dynamic relaxation (DR) method, including the minimum residual force (MRF) scheme, the minimum residual energy (MRE) scheme, the minimum displacement increment (MDI) scheme, and the minimum kinetic energy (MKE) scheme, are commonly used to capture snapping phenomena in post-buckling structural analysis. Analogous to DR incorporating arc-length constraints, these combined numerical schemes effectively handle snap-through problems, but their capability to address snap-back behavior remains largely unverified. This study aims to investigate the numerical stability of the DR-MRF, DR-MRE, DR-MDI, and DR-MKE methods. Unexpectedly, all these methods are found to be unconditionally unstable and inadequate for snap-back problems. Key findings include (1) the establishment of a general finite-difference equation in canonical form for these four methods, with numerical stability governed by the spectral radius of the recursive amplification matrix; and (2) the discovery of a special property of the tangent stiffness matrix, where a specific minor becomes negative during snap-back, causing the spectral radius to exceed one and leading to instability. These conclusions are supported by numerical verifications on nonlinear springs, trusses, and frames. Simulations consistently demonstrate the reliability of these findings across different structural configurations.