A posteriori local error estimation and adaptive time‐stepping for newmark integration in dynamic analysis

Abstract

SUMMARYA simple u posteriori local error estimate for Newmark time integration schemes in dynamic analysis is presented, based on the concept of a so called 'post-processing' technique. In conjunction with the error estimate, an adaptive timestepping algorithm is described, which adjusts the time step size so that the local error of each time step is within a prescribed error tolerance. Numerical examples given in the paper indicate that the error estimate is asymptotically convergent, computationally efficient and convenient, and the adaptive time-stepping scheme can predict a nearly optimal step size from time to time, thus making the numerical solution reliable in an efficient manner.

Asymptotic upper bounds for the errors of Richardson extrapolation with practical application in approximate computations

“…To briefly review the general process of time integration analysis, in view of Figure 3, sizes of time steps or a criterion for adaptive time stepping, e.g. see [47], should be set, first. Then, the main analysis starts from the initial conditions (if needed, after applying a starting procedure [46]); and with marching along the time axis, approximate responses are being determined at discrete time stations, consecutively, based on the formulation of the integration method, and, if needed, some nonlinearity considerations.…”

A technique for time integration analysis with steps larger than the excitation steps

“…Efforts for relaxing Equations (5) and implementing larger t are reported in the literature, e.g. adaptive time stepping [20,40] and commenting the implementation of unconditionally stable time integration methods (h = ∞), see [19,41]. Nevertheless, in real problems, convergence and accuracy are not necessarily the only restrictions affecting the selection of the t. An additional restriction is the step size by which some excitations are recorded [8,9,37]; not all excitations are available analytically, e.g.…”

“…To briefly review the process of time integration, the sizes of time steps or a criterion for adaptive time stepping, e.g. see [20], should be set before any computation. Then, the analysis starts from the initial conditions (if needed, after applying a starting procedure [21]), and with marching along the time axis (see Figure 1), approximate responses are being determined at discrete time stations, sequentially, based on the formulation of the integration method.…”

“…Equation (20) can also typically be considered valid for e 1 i . From Equations (17) and (18) and the general structure of the LP vector, e.g.…”

Numerical simulation of 3D liquefaction disasters using an automatic time stepping method

“…From the previous studies, the earliest and practical automatic stepping procedure for finite element method is proposed by Zienkiewicz and Xie (1991). Then, Zeng et al (1992) modified Zienkiewicz's method by a new posteriori local error estimator. At present, the automatic time stepping method is widely used in structural and fluid engineering (Hulbert and Jang 1995;Jackson 2012;Tang et al 2008).…”