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 g...

[1]  Peter A. Cook,et al.  Nonlinear dynamical systems , 1986 .

[2]  K. Bathe,et al.  On a composite implicit time integration procedure for nonlinear dynamics , 2005 .

[3]  Nathan M. Newmark,et al.  A Method of Computation for Structural Dynamics , 1959 .

[4]  K. Bathe.  Conserving energy and momentum in nonlinear dynamics: A simple implicit time integration scheme , 2007 .

[5]  Augusto Beléndez,et al.  Exact solution for the nonlinear pendulum , 2007 .

[6]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[7]  James M. Ricles,et al.  Stability Analysis of Direct Integration Algorithms Applied to MDOF Nonlinear Structural Dynamics , 2008 .

[8]  Thomas J. R. Hughes,et al.  Stability, convergence and growth and decay of energy of the average acceleration method in nonlinear structural dynamics , 1976 .

[9]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[10]  Pierre Pegon,et al.  α-Operator splitting time integration technique for pseudodynamic testing error propagation analysis , 1997 .

[11]  Randolph E. Bank,et al.  Transient simulation of silicon devices and circuits , 1985, IEEE Transactions on Electron Devices.

[12]  Thomas J. R. Hughes,et al.  Implicit-explicit finite elements in nonlinear transient analysis , 1979 .

[13]  K. Bathe.  Finite Element Procedures , 1995 .

[14]  K. Bathe,et al.  Stability and accuracy analysis of direct integration methods , 1972 .

[15]  Anil K. Chopra,et al.  Dynamics of Structures: Theory and Applications to Earthquake Engineering , 1995 .

[16]  W. Haddad,et al.  Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach , 2008 .