Towards optimal use of the explicit β1/β2-Bathe time integration method for linear and nonlinear dynamics

Abstract

In an earlier publication, we proposed a new explicit time integration scheme, the ${\beta }_1/{\beta }_2$-Bathe method, which is simple in its formulation and showed remarkable accuracy in the solution of problems [1]. A particular strength of the method is that it can directly be used as a first-order or second-order scheme by a change of the values of ${\beta }_1$ and ${\beta }_2$. While good results are obtained with reasonable values of ${\beta }_1$ and ${\beta }_2$, for excellent accuracy better values of the parameters need to be chosen. We propose in this paper values of ${\beta }_1$ and ${\beta }_2$ for the first-order scheme, best used in wave propagation analyses, and separate values for the second-order scheme, best used in analyses of structural vibrations. In each case, one set of values of $({\beta }_1,{\beta }_2)$ is given and to possibly improve the results only one of the parameters needs to be changed, that is, ${\beta }_1$ for wave propagations and ${\beta }_2$ for structural vibrations, making the scheme a one-parameter method. Another strength of the procedure is that physical damping can directly be included in the solution, the effect of which on the stability and accuracy of the solutions we analyze in the paper. The use of the solution scheme in nonlinear analysis is, as we show in the paper, a simple extension from linear analysis. Finally, we give various solutions using the explicit ${\beta }_1/{\beta }_2$-Bathe method in linear and nonlinear analyses to illustrate the performance of the method with the given recommendations for its use.