2. Benchmark solution

2.1. Calculation method

With this test case we want to study the behavior of the various integration schemes in implicit times of the operator DYNA_NON_LINE. It is therefore not a question of trying to reproduce an analytical solution as faithfully as possible.

We therefore choose a time step \(\mathit{dt}\mathrm{=}{10}^{\mathrm{-}2}s\), which is sufficiently small compared to the natural pulsation of the system, and we will solve the linear transient problem with the DYNA_NON_LINE operator.

For the non-dissipative diagram of the mean acceleration (keyword NEWMARK) it would be possible to calculate the analytical solution to compare with it. The resolution is tested while moving or accelerating, which must obviously give the same results.

Figure HHT uses a ALPHA coefficient =-0.3.

For the other schemes that one wishes to test and which are dissipative, obtaining an analytical solution is not easy.

We therefore chose to compare all the calculations with a numerical solution obtained with the Matlab code. For this, the various diagrams were programmed in Matlab.

We will therefore perform several transition calculations in a single step: with the mean acceleration scheme, the modified mean acceleration scheme, the complete HHT diagram and the Krenk diagram. For this last dissipative schemes (with the values of the parameters chosen), the resolution in displacement as well as in speed is validated.

Then we test the calculation again with the complete diagram HHT, to validate the tracking mechanism with this diagram (we do two pursuits, the first at \(\mathrm{0,2}s\) and the second at \(\mathrm{0,35}s\)).

2.2. Reference quantities and results

The comparisons will focus on the displacement and acceleration of the point mass \(M\) at the following times: \(\mathrm{0,5}s\), \(\mathrm{0,7}s\), and \(1s\).