2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The reference values of the natural frequencies are obtained numerically: the results obtained from a Fourier analysis and two 3D analyses are averaged. The formulation for the fluid is the same in all three cases: it is an incompressible fluid, discretized by a displacement potential. The Fourier shell is of the Reissner type, the 3D shell models are respectively of the Mindlin (3D-1) and hybrid (3D-2) type. The analyses were carried out using the SAMCEF software. In each case, the meshes are refined until a stable solution is obtained.
In order to improve the identification of these modes, SAMCEF identifies the standardized degree of freedom and imposes the same in the tests. It will thus be possible to compare the generalized masses.
2.2. Benchmark results#
The table below provides the reference values for the first mode of harmonics 2 to 7 (the harmonic « n » is in \(\mathrm{cos}(n\theta )\)). The relative deviation is calculated with respect to the numerical result that is farthest from the reference.
Harmonic |
Fourier (\(\mathit{Hz}\)) |
Model 3D-1 (\(\mathit{Hz}\)) |
Model 3D-2 (\(\mathit{Hz}\)) |
Reference (\(\mathit{Hz}\)) |
Difference (%) |
|
\(4\) |
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\(3\) |
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\(5\) |
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\(2\) |
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\(6\) |
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\(7\) |
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2.3. Uncertainty about the solution#
Uncertainty varies from frequency to frequency. The largest relative difference is less than 3% (see table above).