Accuracy and stability of a Proposed Implicit Time Integration Method (ζ-Method) Based on a Sinusoidal Interpolation Function for Acceleration

Abstract

In this paper, the accuracy and stability of an implicit numerical method (ζ-method) is investigated. It is shown that ζ-method presents high accuracy and efficiency for the dynamic response analysis by assuming a sinusoidal interpolation function for acceleration between two successive time steps. Assuming a sinusoidal distribution of acceleration results in similar types of equations for velocity and displacement since the integration of a sine term contains sine and cosine terms. For this method, a parameter (denoted as ζ) is used as the frequency of the sinusoidal interpolation function which significantly affects the accuracy and stability of the method. The equations and derivations are presented in detail and the best value for ζ is obtained through multi-objective optimization procedures to minimize the errors. The accuracy and stability of the method have been investigated in terms of period elongation, amplitude decay, and spectral radius. Finally, the method has been evaluated by several numerical examples (linear and nonlinear SDOF, and linear MDOF). In some examples, it was observed that the ζ-method yielded better results than other numerical methods. Moreover, an interpolated version of the method was introduced which was more accurate in comparison with similar methods with equal execution time.