Locating Events in Step‐by‐Step Integration of the Equations of Motion

Abstract

The calculation of time‐step sizes to prevent interstep events in multilinear problems is addressed. The treatment is restricted to the Newmark‐Beta family of implicit integration methods. Closed‐form equations in the fractional steps are derived for single‐degree‐of‐freedom (SDOF) systems. Explicit solutions are, however, not feasible because the equations are cubic. Numerical results indicate that the location of events can lead to marked improvements in accuracy, when compared to solutions that handle the nonlinear error at the ends of the time steps. It is contended that the derivation of closed‐form expressions for the substep sizes, in general multidegree‐of‐freedom (MDOF) systems, is precluded by coupling. Identification of the steps that contain events is prerequisite to any event location technique. In this regard, it is shown that a customary test to detect unloading, based on incremental hinge rotations, sometimes fails. An approach that does not suffer from the identified shortcoming is presented. The numerical stability characteristics of an approximate event location strategy based on interpolation is examined using an energy approach. It is shown that the constant‐average‐acceleration method does not satisfy the requirements for unconditional stability when the events are interpolated.