2. Objective of the test case, validation#

The objective of the test case is to validate the functionality SPEC_CORR_CONV_3de the DEFI_SPEC_TURB operator, which makes it possible to define a spectrum using analytical functions of the space and frequency variable.

In modeling A, the spectrum defined is purely theoretical, since the functions are sines. The calculation of the double integral \({\mathrm{\int }}_{\Omega }{\mathrm{\int }}_{\Omega }{\Phi }_{i}(\underline{{x}_{1}})\text{.}{S}_{f}(\underline{{x}_{1}},\underline{{x}_{2}},\omega )\text{.}{\Phi }_{j}(\underline{{x}_{2}})d{\Omega }_{1}d{\Omega }_{2}\) can therefore be solved analytically, assuming that the first 2 eigenmodes of the beam are also sine functions. In modeling B, the spectrum defined is representative of a flow downstream of a mixing grid on a fuel rod (Corcos model).

2.1. Conduct of the test case#

  • Eigenmode calculation: the first two eigenmodes are calculated, they are of the form \(\phi (y)\mathrm{=}\mathrm{sin}(\pi y)\) and \(\phi (y)\mathrm{=}\mathrm{sin}(2\pi y)\),

  • definition of the functions of the turbulent spectrum:

    • A \({S}_{\mathit{xx}}\mathrm{=}f\text{.}\mathrm{sin}(\pi {y}_{1})\text{.}\mathrm{sin}(\pi {y}_{2})\) modeling

    • B modeling

\({S}_{f}(\underline{{x}_{1}},\underline{{x}_{2}},\omega )\mathrm{=}\mathrm{\{}\begin{array}{c}{S}_{\mathit{xx}}\mathrm{=}\text{exp}(\mathrm{-}\frac{\mathrm{\mid }{y}_{2}\mathrm{-}{y}_{1}\mathrm{\mid }}{{\lambda }_{\text{cx}}(\omega )})\text{.}\text{exp}(\mathit{j\omega }\frac{{y}_{2}\mathrm{-}{y}_{1}}{{U}_{c}}){S}_{f}({y}_{1},{y}_{1},\omega )\\ {S}_{\mathit{yy}}\mathrm{=}\text{exp}(\mathrm{-}\frac{\mathrm{\mid }{y}_{2}\mathrm{-}{y}_{1}\mathrm{\mid }}{{\lambda }_{\text{cy}}(\omega )})\text{.}\text{exp}(\mathit{j\omega }\frac{{y}_{2}\mathrm{-}{y}_{1}}{{U}_{c}}){S}_{f}({y}_{1},{y}_{1},\omega )\end{array}\)

(by adding correlation terms between the efforts according to \(x\) and \(y\) with the functions \({S}_{\mathit{xy}}\) and \({S}_{\mathit{yx}}\)).

  • creation of a table containing the functions associated with the directions,

  • creation of the inter-spectrum with DEFI_SPEC_TURB,

  • projection of the inter-spectrum on the two natural modes calculated with PROJ_SPEC_BASE; the projection is done on the group of elements” BAS “only.

Note:

For B modeling, an excitation is defined in both directions. However, the two calculated modes are in the direction \(X\) * (the direction \(\mathit{DZ}\) is blocked). So the arousal according to \(Y\) will have no influence on the result of the modal arousal.

2.2. Validation of results#

2.2.1. Modeling A#

For modeling A, the validation is analytical. In fact, the integrals to be calculated are as follows:

  • auto-spectrum mode 1: \({\mathrm{\int }}_{0}^{0.5}{\mathrm{\int }}_{0}^{0.5}{\mathrm{sin}}^{2}(\pi {y}_{1}){\mathrm{sin}}^{2}(\pi {y}_{2}){\mathit{dy}}_{1}{\mathit{dy}}_{2}\mathrm{=}\frac{1}{4}\text{.}\frac{1}{4}\mathrm{=}0.0625\)

  • inter-spectrum mode 1 — mode 2: \({\mathrm{\int }}_{0}^{0.5}{\mathrm{\int }}_{0}^{0.5}\mathrm{-}{\mathrm{sin}}^{2}(\pi {y}_{1})\mathrm{sin}(\pi {y}_{2})\mathrm{sin}(2\pi {y}_{2}){\mathit{dy}}_{1}{\mathit{dy}}_{2}\mathrm{=}\mathrm{-}\frac{2}{3\pi }\text{.}\frac{1}{4}\mathrm{=}\mathrm{-}0.05305\)

  • auto-spectrum mode 2: \({\mathrm{\int }}_{0}^{0.5}{\mathrm{\int }}_{0}^{0.5}\mathrm{sin}(\pi {y}_{1})\mathrm{sin}(2\pi {y}_{1})\mathrm{sin}(\pi {y}_{2})\mathrm{sin}(2\pi {y}_{2}){\mathit{dy}}_{1}{\mathit{dy}}_{2}\mathrm{=}\mathrm{-}\frac{2}{3\pi }\text{.}\mathrm{-}\frac{2}{3\pi }\mathrm{=}0.04503\)

2.2.2. B modeling#

For B modeling, validation is done by non-regression.

  • auto-spectrum mode 1: \(0.11\)

  • inter-spectrum mode 1 — mode 2: \(0.11+\mathrm{0.01534j}\)

  • auto-spectrum mode 2: \(0.11\).