3. Modeling A#

Concrete slabs are modelled in Q4GG elements, on quadrangular or triangular meshes. The columns are modelled in elements of type POU_D_E. The ground springs are, for their part, of the K_ TR_D_N type.

3.1. Analysis of the results#

In this chapter, we will trace the vertical changes in the displacement and acceleration of point \(\mathit{PP2}\) (in the middle of the upper slab). Each time, we will superimpose the different answers obtained:

  1. resolution with CALC_EUROPLEXUS then continuation with DYNA_NON_LINE explicitly,

  2. resolution with CALC_EUROPLEXUS over the entire study interval,

  3. resolution with explicit DYNA_NON_LINE over the entire study interval,

  4. switch CALC_EUROPLEXUS to DYNA_NON_LINE implicit,

  5. Switch DYNA_NON_LINE explicit to implicit.

It will thus be possible to quantify:

  • the precision when rereading data MED between the codes and the good agreement of the models (curve 1),

the differences in the model between Europlexus and*Code_Aster (curves 2 and 3) with the same time diagram,

  • the differences between explicit solution and solution with the schema toggle (curves 4 and 5).

Analysis on accelerations should amplify the differences, compared to the differences in calculated displacements.

For the following graphs, we deliberately increased the simulated time period, in order to have a more relevant analysis. In the test case command file, this simulation time is much shorter to reduce time CPU.

3.1.1. Comparison of calculated trips#

_images/1000000000000AD6000008C8931F28176B514BEC.jpg

The response is of an oscillating type and the differences are small. We just observe a very slight phase difference between explicit and implicit responses, which is due to time patterns. The difference in amplitude is negligible.

In order to complete the analysis, we will zoom in on the tipping point at time 0.0075 s:

_images/1000000000000AD1000008C833973B321D7CE09A.jpg

The flip flop generates a slight disturbance, but the responses remain the same, whether you go from Europlexus to*Code_Aster* (black curve) or whether all the resolution is done in Code_Aster (brown curve).

It is important to note that this difference due to the switch is not getting bigger and that the answers remain very close to the reference solutions (without a switch).

3.1.2. Comparison of calculated accelerations#

_images/1000000000000AC20000089BA7D8DDCB199F78B5.jpg

Accelerations are much more hectic than trips, but the differences between digital solutions remain moderate. More exactly, we can clearly see the influence of the implicit time diagram that smoothes the response (dissipative diagram type HHT).

_images/1000000000000AB20000089BF83A2BF179597E44.jpg

By zooming in on the tipping point, significant disturbances in acceleration appear. These numerical oscillations cannot be completely annihilated, even by changing the parameters of the schema switching method (choice of the schema during the balancing phase, parameter of this schema, choice of the schema before and after the changeover…). In fact, we have been able to verify that these oscillations are very largely caused by the change in mass matrix: in fact, explicitly we use a lumped mass matrix, while implicitly it is the consistent mass matrix that is chosen. During the switch, we suddenly pass from one mass matrix to the other and this disrupts the solution. The rebalancing strategy does not completely manage to erase this jump.

A workaround could be to switch progressively from one mass matrix to another, using a variable mass matrix over a few steps of time, which would be a linear combination of the two matrices (lumped and consistent). The interest of this algorithmic evolution will remain to be quantified on this test case, for example.

3.2. Tested values#

We will test the values obtained (displacements and accelerations) at the final moment: \(\mathrm{0,007}s\).

Each value tested will in fact be the absolute value of the relative difference between the result to be tested and the reference solution under consideration. This formula should tend towards 0 and it is directly a relative value, which explains why in the TEST_FONCTION operator we specify CRITERE =” ABSOLU “because, without that, we would try to make the value relative a second time.

In practice, these tested values can only tend to 0 if the difference between the same calculation carried out with Code_Aster or Europlexus is zero, which is not the case. More exactly, the differences for the solutions with time switch cannot be less than the differences between the two reference solutions (complete calculation with Europlexus and complete calculation with Code_Aster explicitly).

We therefore start by measuring the relative differences between the two reference solutions. On the go, we have: \(\mathrm{2,656188}{.10}^{-3}\) and in acceleration: \(\mathrm{0,016270437}\)

Next, we will analyze the relative differences induced by the resumption of the calculation between Europlexus and Code_Aster (explicit, therefore without a switch). On the move, we have: \(\mathrm{4,93101}{.10}^{-04}\) and in acceleration: \(\mathrm{0,0380}\).

The differences induced by the time switch should be of the same order of magnitude. Indeed, the toggle cannot correct these differences inherent in the differences between the codes.

We start by comparing the solution with switching Europlexus to implicit Code_Aster. When moving, we have a relative difference of: \(\mathrm{5,02195}{.10}^{\mathrm{-}04}\) and in acceleration: \(\mathrm{0,019848325}\).

Finally, we give the relative differences with the solution obtained by switching from explicit to implicit, but always staying with*Code_Aster*. When moving, we have a relative difference of: \(\mathrm{0,011460178}\) and in acceleration: \(\mathrm{0,045500117}\).

We note that the differences in accelerations are greater than those in the movements, which is logical because the movements are more regular quantities than the accelerations, as can be verified on the graphs in paragraphs 3.1.1 and 3.1.2.

Next, we note that on movements, the rocker introduces very little disturbance, while on accelerations, oscillations, admittedly damped, appear. After analysis, it turns out that they are mainly due to the sudden transition from a lumped mass matrix to a consistent matrix.

Regarding the calculation of SRO, we will calculate the maximum relative difference (over the entire frequency range and for an equivalent damping of 5%) with the reference solution, whether for the SRO obtained from the OBSERVATIONet of the ARCHIVAGE. These relative differences are of the order of 5 to 8% and are due to the switch and to the differences in the time steps where the SRO input data are archived.