Objective of the test case, validation
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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 :math:`{\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).


Conduct of the test case
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* Eigenmode calculation: the first two eigenmodes are calculated, they are of the form :math:`\phi (y)\mathrm{=}\mathrm{sin}(\pi y)` and :math:`\phi (y)\mathrm{=}\mathrm{sin}(2\pi y)`,


* definition of the functions of the turbulent spectrum:


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


    * B modeling

:math:`{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 :math:`x` and :math:`y` with the functions :math:`{S}_{\mathit{xy}}` and :math:`{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* :math:`X` * *(the direction* :math:`\mathit{DZ}` *is blocked). So the arousal according to* :math:`Y` *will have no influence on the result of the modal arousal.*


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


* auto-spectrum mode 1: :math:`{\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: :math:`{\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: :math:`{\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`


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


* auto-spectrum mode 1: :math:`0.11`


* inter-spectrum mode 1 — mode 2: :math:`0.11+\mathrm{0.01534j}`


* auto-spectrum mode 2: :math:`0.11`.