For direct time integrations: A comparison of the Newmark and p_∞-Bathe schemes

摘要

We consider the unconditionally stable Newmark and rho(infinity)-Bathe methods for the direct time integration of the finite element equations in structural dynamics and wave propagations. In our evaluation of the Newmark method we consider the parameters delta and alpha, and in the rho(infinity)-Bathe method we consider the parameters gamma and rho(infinity), with 0 < gamma < infinity, gamma not equal 1 and rho(infinity) is an element of [-1;+1]. We show that the Newmark method as usually used with its delta and alpha parameters, alpha = 0.25(delta + 0.5)(2) and delta >= 0.5, is a special case of the rho(infinity)-Bathe method. We also show that the beta 1/beta 2-Bathe method is a special case of the rho(infinity)-Bathe scheme. The study of the curves of numerical dissipation and dispersion shows that the rho(infinity)-Bathe method provides effective dissipation and dispersion whereas the Newmark method lacks in that regard. To illustrate our theoretical findings we give the results of some example solutions of structural dynamics and wave propagations. Our study also shows that further research is needed to identify the optimal use of the rho(infinity)-Bathe scheme and other implicit methods in wave propagation analyses. (C) 2019 Elsevier Ltd. All rights reserved.