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Lyapunov Stability and Accuracy of Direct Integration Algorithms Applied to Nonlinear Dynamic Problems
AbstractIn structural dynamics, direct integration algorithms are commonly used to solve the temporally discretized differential equations of motion. Numerous research efforts focused on the stability and accuracy of different integration algorithms for linear elastic structures. However, investigations of those properties applied to nonlinear structures are limited. Systematic Lyapunov stability and accuracy analyses of several direct integration algorithms for nonlinear structural dynamics are presented. These integration algorithms include the implicit methods of the Newmark family of integration algorithm and the TRBDF2 algorithm, and the explicit methods of operator-splitting (OS) algorithms. Two versions of the OS algorithms, using initial and tangent stiffness formulations, are investigated. The latter one is shown to possess similar stability properties to the implicit Newmark integration. Some arguments of stability regarding these direct integration algorithms from past studies are found to be groundless. An approach that enables performing the stability analysis numerically is proposed. It transforms the stability analysis to a problem of convex optimization. In addition, accuracy of the previously discussed five integration algorithms is investigated using a geometrically nonlinear test problem, in which acceptable amounts of period elongation and amplitude decay were evident.
Lyapunov Stability and Accuracy of Direct Integration Algorithms Applied to Nonlinear Dynamic Problems
AbstractIn structural dynamics, direct integration algorithms are commonly used to solve the temporally discretized differential equations of motion. Numerous research efforts focused on the stability and accuracy of different integration algorithms for linear elastic structures. However, investigations of those properties applied to nonlinear structures are limited. Systematic Lyapunov stability and accuracy analyses of several direct integration algorithms for nonlinear structural dynamics are presented. These integration algorithms include the implicit methods of the Newmark family of integration algorithm and the TRBDF2 algorithm, and the explicit methods of operator-splitting (OS) algorithms. Two versions of the OS algorithms, using initial and tangent stiffness formulations, are investigated. The latter one is shown to possess similar stability properties to the implicit Newmark integration. Some arguments of stability regarding these direct integration algorithms from past studies are found to be groundless. An approach that enables performing the stability analysis numerically is proposed. It transforms the stability analysis to a problem of convex optimization. In addition, accuracy of the previously discussed five integration algorithms is investigated using a geometrically nonlinear test problem, in which acceptable amounts of period elongation and amplitude decay were evident.
Lyapunov Stability and Accuracy of Direct Integration Algorithms Applied to Nonlinear Dynamic Problems
Liang, Xiao (author) / Mosalam, Khalid M
2016
Article (Journal)
English