On practical integration of semi-discretized nonlinear equations of motion. Part 1: reasons for probable instability and improper convergence

Time integration is the most versatile method for analyzing the general case of nonlinear semi-discretized equations of motion. However, the approximate responses of such analyses generally do not converge properly, and might even display numerical instability. This is a very significant shortcoming especially in practical time integration. Herein, after illustrating that this phenomenon is viable even for very simple nonlinear dynamic models, sources of the shortcoming are discussed in detail. The conclusion is that in time integration of nonlinear dynamic mathematical models of physically stable structural systems, responses may converge improperly for three major reasons. These reasons are: (1) inadequate number of iterations before terminating nonlinearity solutions; (2) deficiencies in the formulation of some time integration methods; and (3) the inherent behaviour of the models of some special dynamic systems. In addition, limitations on computational facilities and improper consideration of these limitations may impair the numerical stability and convergence of the computed responses. The differences between static and dynamic analyses are also discussed from the viewpoint of the numerical errors induced by nonlinearity.