Insight to the Newmark Implicit Time Integration Method for Solving the Wave Propagation Problems

Newmark implicit time integration method is one of the oldest and most powerful methods used for dynamic analysis of structures and wave propagation problems. Recently, researchers have proposed a straightforward time integration method to analyze wave propagation problems. In a series of research papers, new time integration methods were developed. They showed that these methods are more capable than the Newmark implicit time integration method in solving wave propagation problems. In this paper, a state of the Newmark method for analyzing the wave propagation problems is presented. The results are compared with Bathe family methods, including Noh-Bathe, standard Bathe, β1/β2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta_{1} /\beta_{2}$$\end{document}-Bathe and ρ∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho_{\infty }$$\end{document}-Bathe methods. For this purpose, firstly, a numerical dispersion analysis is presented to evaluate the Newmark method in different ways. Then, the Bathe family methods in evaluating the response of wave propagation problems are compared with results from the Newmark method. The numerical evaluation is performed for two problems. Results indicate that this type of Newmark method has better performance than the Newmark trapezoidal and several Bathe family methods.