An explicit integration method with third-order accuracy for linear and nonlinear dynamic systems

Abstract

Computational efficiency and accuracy are two of the most important properties of integration methods. In this study, an explicit single-step integration method with third-order accuracy is proposed to improve computational efficiency and accuracy. The proposed method can achieve fourth-order accuracy in terms of displacement, velocity, and acceleration responses without physical damping. The algorithmic dissipation property is also improved to filter out high-frequency spurious responses on the premise of sufficient accuracy in the low-frequency response domain. Moreover, since the proposed method can advance the calculation step-by-step via the known displacement and velocity items, the time-consuming factorization of an effective stiffness matrix is not required in the calculation for the structure with a lumped mass matrix. In other words, high computational efficiency is ensured in the proposed method. The properties of the proposed method (i.e., the accuracy, convergence, dissipation, efficiency, and effectiveness of the nonlinearity) are demonstrated by using four representative examples. Specifically, a single-degree-of-freedom (SDOF) dynamic system with a theoretical solution is adopted to comparatively evaluate the accuracy and convergence of the proposed method, a typical nonlinear dynamic system is used to investigate the effectiveness of the nonlinear calculation; a linear Howe truss model is employed to explore the effectiveness and efficiency in the calculation of a multi-DOF structure, and a nonlinear wellbore modelis used to demonstrate the potential to solve the large-scale nonlinear system. Results show that the proposed method can obtain more accurate results in the nonlinear dynamic calculations; and its time consumption is around 65% of that of the Yuan method, 38% of that of the Kim method, and 30% of that of the Zhai method.