2. Benchmark solution

2.1. Calculation method used for the reference solution

The only results from the literature [bib1] are of modal types: natural frequencies and the shapes of certain modes.

Given the need to test the operator DYNA_NON_LINE, and given the relative complexity of the 3D model, it is not possible to find the natural frequencies by transient analysis in a reasonable CPU time. Linear search is also used.

For information, this type of analysis conducted with a random loading corresponding to white noise requires, for reasons of probabilistic convergence, a calculation for a physical loading time of \(250s\), which corresponds to a CPU time of a few hours.

In order to have a calculation time of the order of a few minutes, it is mandatory to calculate the evolution over a short time (a few seconds). This restrictive framework does not make it possible to retrieve the modal analysis results precisely and in a manner compatible with automated post-processing.

The validation provided by this test can therefore only be of the non-regression type of numerical solution.

As the fluid-structure coupled calculation functionalities are already the subject of a certain number of validation tests elsewhere, this limitation to non-regression for this particular test case is not a prohibitive step.

As additional validation, the complete calculation with signal of \(250s\) was carried out. The spectra at the observation points showed good agreement with the modal analysis results of [bib1].

To validate the stability analysis on this fluid-structure problem, we will use the CRIT_STAB keyword from DYNA_NON_LINE.

2.2. Benchmark results

Displacement values are tested at different times, in the \(x\) direction, for two points in the mesh: \(\mathrm{N145}\) and \(\mathrm{N3119}\). These points are on the free surface, on either side of the deformable wall, as can be seen in the diagram in paragraph [§1.1].

As for stability, as we are not going to use the geometric stiffness matrix (which is unavailable for elements DKT used here), the analysis can only be of the type of search for singularities in the stiffness matrix (therefore an eigenvalue that tends to 0).

Since the problem is linear elastic, we do not expect, on the one hand, no instability and on the other hand, we should find the critical eigenvalue (or critical load) only for the same model but without the fluid and which is equal to -2.47726. This eigenvalue will be the same at each calculation step because the problem remains linear.

2.3. Uncertainty about the solution

Numerical solution (calculated with version 7.3.6 of the code).

2.4. Bibliographical reference

1. BERMUDEZ A., RODRIGUEZ R., SANTAMARINA D.: « Finite element computation of sloshing modes in containers with elastic baffle plates », Int. Mr Numer. Meth. In Engrg., Vol. 56, 447-467, 2003