A Priori Error Estimator of the Generalized-α Method for Structural Dynamics

Abstract

Time integration algorithms are widely used in structural dynamic problems to compute time responses from spatially discretized equations that are commonly obtained by the finite element methods. When integrating the equations in the time domain, two conflicting factors should be considered: accuracy and cost. These conflicting factors depend largely on the selection of the time step sizes. In the last decade, much research has been carried out for time stepping algorithms to automatically determine the optimal time step size that maximizes accuracy while minimizing computational expenses. In many cases, time stepping algorithms have aimed at finding the largest possible time step size, while maintaining prescribed accuracy.

Hulbert and Jang1 developed a posteriori local error estimator and time step control algorithm for the generalized- method2. However, this time stepping algorithm uses a posteriori error estimator, which have a feedback process to determine the next time step size. In this case, the time stepping procedure may be described as the following. (1) Compute a temporary acceleration with a temporary time step size, (2) compute an error estimator with the temporary acceleration, (3) determine the true next time step size with the error estimator and (4) compute the acceleration again with the next time step size. If this feedback process is avoided, much computational cost can be saved.

In this study, a priori error estimator is developed to solve structural dynamic problems, based on the generalized- method. Since the proposed error estimator is computed with only information in the previous and current time steps, the time step size can be adaptively selected without a feedback process, which is required in most conventional posteriori error estimators. This study shows that the automatic time stepping algorithm using the priori estimator performs more efficient time integration, when compared to algorithms using the posteriori estimator. In particular, the proposed error estimator can be usefully applied to large-scale structural dynamic problems, because it is helpful to save computation time. To verify efficiency of the algorithm, several examples are numerically investigated.