The Bathe time integration method in wave propagations

Several papers have been published about the Bathe method in wave propagation, the most recent is published by S-B. Kwon, K-J. Bathe, and G. Noh: An analysis of implicit time integration schemes for wave propagations.

Some years ago we set up a small model to test the method and compared with the Newmark method, see (scroll down to the pipe model) http://www.adina.com/newsgH101.shtml

A similar model has been set up again, with only a linear (acoustic) fluid. The problem has also been set up in three other codes, that use the HHT and Generalized-alpha methods. We know that other researchers have solved similar problems in other finite element codes, but this is the first time that we have investigated our selves the set up of a problem and the numerical outcome.

For transient fidelity, the numerical dissipation is turned off. Because mass acceleration is used as boundary condition in two codes, the velocity derivative has been used, which requires special attention. Here the triangular mass flux (and hence wave pressure) cannot be followed through just derivation of the velocity to the mass acceleration because of the discontinuous functions.

What is also shown by this simple example is that the Newmark method (and HHT and Gen-alpha methods) produce dispersion (frequency dependent wave propagation speed) and spreads the pressure peaks and valleys. Which is a loss of fidelity. In our paper On blowdown analysis with efficient and reliable direct time integration methods for wave propagation and fluid-structure-interaction response we could not show any apparent advantages by using the Bathe method for the linear problem, because the solution time was longer than the Newmark method and the accuracy seemed similar for the full vibroacoustic problem. However, we now know that higher fidelity in the wave propagation is expected in all cases with the Bathe method. For the non-linear problems the Bathe method is stable and produce highly accurate results.

The Bathe method was originally developed for non-linear direct integration dynamic problems, but as shown it provides unusual characteristics also in linear dynamics such as wave propagation with low dispersion.

The ADINA input and plot files can be sent, if some one wants to test by themselves. The input files to the other codes are not distributed. However, the problem is very simple and following the ADINA input file, experienced analysts should easily be able to reproduce the results in the code they are familiar with.