An Adaptively Filtered Precise Integration Method Considering Perturbation for Stochastic Dynamics Problems

The difficulty in solving stochastic dynamics problems lies in the need for a large number of repeated computations of deterministic dynamic equations, which has been a challenge in stochastic dynamics analysis and was discussed in this study. To efficiently and accurately compute the exponential of the dynamics state matrix and the matrix functions due to external loads, an adaptively filtered precise integration method was proposed, which inherits the high precision of the precise integration method, improves the computational efficiency and saves the memory required. Moreover, the perturbation method was introduced to avoid repeated computations of matrix exponential and terms due to external loads. Based on the filtering and perturbation techniques, an adaptively filtered precise integration method considering perturbation for stochastic dynamics problems was developed. Two numerical experiments, including a model of phononic crystal and a bridge model considering random parameters, were performed to test the performance of the proposed method in terms of accuracy and efficiency. Numerical results show that the accuracy and efficiency of the proposed method are better than those of the existing precise integration method, the Newmark-β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document} method and the Wilson-θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document} method.