Rho_inf Bathe: Low pass filtering

The rho_inf Bathe method is a composite time integration method, with adjustable numerical dissipation at the high frequency limit.

Effectively this is analog to a low pass filter. The low frequency oscillations are adequately resolved with the selected time step for the problem, but the higher frequencies are not resolved. When a structural dynamic problem is to be solved the time step is selected so that the cut-off frequency is resolved with 20 time steps per period. This means that all frequencies below the cut-off frequency are resolved and of high accuracy, but higher frequencies are not.

One order of magnitude higher than the cut-off frequency the temporal resolution is just 2 time steps per period, obviously insufficient. However, in finite element analyses the number of eigenfrequencies of the mesh is the same as the number of degrees-of-freedom (N). Hence, a typical model contains thousands or millions of eigenfrequencies.

It is therefore common to solve transient structural dynamic problems with a mode superposition, where to eigenmodes are calculated up to a cut-off frequency that is is governed by the frequency content in the loading, this means that only the lowest eigenmodes are involved, where M << N.

When the problem is solved by a direct dynamic method, this reduction to participating oscillations are not done. One may use time integration methods that are unconditionally stable (such as the trapezoidal rule), but that also means that the high frequency oscillations are not accurate. It is therefore advantageous to use numerical dissipation to filter out the high accuracy resolved lower frequency oscillations, as with the mode superposition method. This can be done with the Bathe methods available in ADINA.

The rho_inf method implemented in ADINA 9.6 allows the strength of the numerical dissipation to be adjusted, the standard Bathe method always gives the maximum dissipation. Hence, the selected time step is used to control the cut-off frequency of the solution and the rho_inf parameter is used to control the strength of the dissipation.

In non-linear structural dynamics, the mode superposition method cannot be used and the unconditional stability of methods in linear direct method is violated, so there the time integration must provide accuracy and stability. This will be further investigated in a separate post.