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A theoretical proof of the invalidity of dynamic relaxation arc-length method for snap-back problems

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A Correction to this article was published on 29 October 2021

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Abstract

Incorporating the arc-length constraint, the dynamic relaxation strategy has been widely used to trace full equilibrium path in the post-buckling analysis of structures. This combined numerical scheme has been shown to be successful for solving snap-through problems, but its applicability to snap-back problems has been rarely investigated and remains unclear. This paper proposes a direct and more general finite-difference equation to investigate the numerical stability of this combined numerical scheme, which is dominated by the spectral radius of amplification matrix. And a key discovery of this paper is that a first minor of the tangent stiffness matrix is always negative once snap back occurs. Due to this negative minor stiffness, the spectral radius is invariably greater than one, resulting in unconditional instability, which demonstrates the invalidity of dynamic relaxation arc-length method for snap-back problems. These important conclusions are corroborated by the numerical results of three representative examples in one-, two- and three-dimensional spaces.

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Abbreviations

\({\mathbf{\rm K}}\) :

Tangent stiffness matrix

\({\mathbf{u}}\) :

Nodal displacement vector

\({\mathbf{f}}\) :

External load vector

\(M_{ji}\), \(M_{ii}\) :

First minor

\({\mathbf{K}}^{*}\) :

Minor stiffness

\({\mathbf{m}}\) :

Fictitious mass matrix

\({\mathbf{c}}\) :

Viscous damping

\({\mathbf{\ddot{u}}}\) :

Acceleration vector

\({\dot{\mathbf{u}}}\) :

Velocity vector

\(\mu\) :

Damping factor

\(\Delta t\) :

Time increment

\(\gamma\) :

Load factor

\({\mathbf{f}}_{{{\text{ref}}}}\) :

Reference load vector

\(l\) :

Arc radius

\(\alpha\) :

Artificial scaled parameter

\({\mathbf{A}}\), \({\mathbf{B}}\) :

Amplification matrix

\({\mathbf{L}}\) :

Load operator

\(\lambda\) :

Eigenvalue

\({\mathbf{P}}\) :

Eigenvectors

\(\rho \left( {\mathbf{A}} \right)\), \(\rho \left( {\mathbf{B}} \right)\) :

Spectral radius

\({\mathbf{I}}\) :

Identity matrix

\({\mathbf{I}}_{0}\) :

Imperfect identity matrix with a zero diagonal element

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Acknowledgements

This research was supported in part by Jiangsu Provincial Entrepreneurship and Innovation Foundation of China (JSEI2017073), the National Natural Science Foundation of China (Grant No. 52008366), and Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ21E080019).

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  1. Spatial Structures Research Center, Department of Civil Engineering, Zhejiang University, Hangzhou, China

    Pengfei Zhang & Chao Yang

  2. MCC Huatian Engineering Technology Co. Ltd., Nanjing, China

    Pengfei Zhang

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Zhang, P., Yang, C. A theoretical proof of the invalidity of dynamic relaxation arc-length method for snap-back problems. Comput Mech 69, 335–344 (2022). https://doi.org/10.1007/s00466-021-02071-9

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