Soroushian A.

Time integration is, at present, the most versatile tool to analyze the nonlinear dynamic behavior of structural systems. The responses computed by time integration are, however, in general, approximations. In presence of nonlinearity, the main algorithmic parameters controlling this approximation (and the corresponding errors) are time step sizes and nonlinearity tolerances. The changes of responses errors with respect to time step sizes is talked about, by times; and recently discussed thoroughly by the author. Nevertheless, the literature seems lacking sufficient, if any, study on the changes of errors with respect to the nonlinearity tolerances of time integration. Instead, it is an accepted practice, probably from static analyses, to consider the implementation of smaller nonlinearity tolerances as a tool to enhance the accuracy in time integration of nonlinear equations of motion. In this paper, after brief theoretical discussions on the concept; by means of simple examples, it is demonstrated that, in time integration analyses, smaller nonlinearity tolerances are not necessarily superior from the point of view of accuracy. The report proceeds with a study on the theoretical reasons of this phenomenon (also comparing static and dynamic cases), later continued with a discussion on the corresponding influences on practical dynamic analyses, and finally, closed with a brief set of conclusions. Some References 1. Belytschko, T., and Schoeberle, D.F. (1975), 'On the unconditional stability of an implicit algorithm for structural dynamics', Journal of Applied Mechanics (E), 42(4), 865-869. 2- Xie, Y.M., and Steven, G.P. (1994), Instability, chaos, and growth and decay of energy of time-stepping schemes for nonlinear dynamic equations, Communications in Numerical Methods in Engineering, 10(5), 393-401. 3- Soroushian, S., Wriggers, P. and Farjoodi, J., (2005), “On practical integration of semi-discretized equations of motion Part 1: reasons for probable instability and improper convergence”, Journal of Sound and Vibration, 284(3-5), 705-731.