9. Summary of results

Modeling A uses fluid \((p,\mathrm{\Phi })\) and heavy free surface elements in formulation \((p,\mathrm{\Phi },z)\), on Hexa8 meshes and quad4 free surface elements. The result obtained reveals a maximum error of 1.51% with respect to the analytical solution of the natural frequencies of the sloshing modes and the acoustic modes.

B modeling involves fluid elements in formulation \((p,\mathrm{\Phi },z)\), on Hexa20 type meshes and free surface elements on quad8 meshes. The results of the natural frequencies of the sloshing modes correspond perfectly with the analytical solution.

Fluid elements in formulation \((p,\mathrm{\Phi },z)\), on cells of the Hexa27 type and of heavy fluid of the Quad9 type are used for modeling C. The results of the natural frequencies of the sloshing modes correspond perfectly with the analytical solution.

D modeling makes it possible to validate fluid elements in formulation \((p,\mathrm{\Phi },z)\), on Penta6-type meshes and free surface elements on Stria3 meshes. The results of the natural frequencies of the sloshing modes reveal a relative error of 1.479% compared to the reference solution.

E-modeling makes it possible to validate Tria6 type free surface elements. They are connected to fluid elements on Penta15 meshes. The results of the natural frequencies of the sloshing modes correspond perfectly with the analytical solution.

The F modeling makes it possible to validate linear acoustic fluid elements (degrees of freedom of pressure \(p\)), on hexa8 meshes. There is a perfect coincidence of the results with the analytical solution (maximum error less than 1.2% for this mesh).

In general, it is observed that the results obtained by elements with quadratic interpolation (HEXA20, HEXA27) are in excellent agreement with the analytical results. Nevertheless, the elements with linear interpolation have an acceptable precision on the results, in accordance with the current recommendations for these meshes in terms of the number of finite element cells per wavelength.