Reference problem
=====================


Geometry
---------


.. image:: images/Cadre2.gif


.. _RefSchema_Cadre2.gif:

Figure 1.1-1: Geometry (not scaled)
We consider a flat metal ring immersed in a fluid medium with an external border for the latter (figure).

In the metal ring, a visco-elastic layer is modeled over a quarter of the circumference.


Material properties
----------------------

For the steel part, an isotropic elastic material is considered:


* Modulus of elasticity: :math:`{E}_{a}=177\mathit{GPa}`


* Poisson's ratio: :math:`{\mathrm{\nu }}_{a}=0.3`


* Density: :math:`{\mathrm{\rho }}_{a}=7450\mathit{kg}\mathrm{.}{m}^{-3}`

     For the fluid, it is water with the following characteristics:


* Speed of sound: :math:`c=1500m\mathrm{.}{s}^{-1}`


* Density: :math:`{\mathrm{\rho }}_{e}=1000\mathit{kg}\mathrm{.}{m}^{-3}`

     For the visco-elastic part, we consider a material of the fractional derivative model type (Zener model) whose shear modulus depends on the pulsation:

.. math::
 :label: eq-1

    {G} ^ {\ text {*}}}\ left (\ omega\ right) =\ frac {{G} _ {0} + {G} _ {\ infty} {(\ mathit {i\ omega\ tau})}}}}} ^ {omega\ tau})}} ^ {\ omega\ tau})} ^ {\ omega\ tau})} ^ {\ omega\ tau})} ^ {\ omega\ tau})} ^ {\ omega\ tau})} ^ {\ omega\ tau})}

The characteristics are as follows:


* Poisson's ratio: :math:`{\mathrm{\nu }}_{v}=0.49` (almost incompressible material)


* Density: :math:`{\mathrm{\rho }}_{v}=1460\mathit{kg}\mathrm{.}{m}^{-3}`


* Coefficient: :math:`{G}_{0}=\mathrm{2,11}\mathit{MPa}`, :math:`{G}_{\mathrm{\infty }}=\mathrm{0,59}\mathit{GPa}`, :math:`\mathrm{\tau }=\mathrm{0,44}\times {10}^{-6}`, and :math:`\mathrm{\alpha }=\mathrm{0,53}`

Note: to improve the calculation of the modes, all these quantities are a-dimensioned in relation to water.

Boundary conditions and loads
-------------------------------------

For the structure part, an embedment condition is imposed on the segment :math:`\mathit{AB}` and a horizontal nodal force :math:`\mathit{Fx}=A\mathrm{,0}\times {10}^{-5}N` is imposed on the point :math:`A`.

A fluid-boundary interaction condition is achieved by the use of 2D_ FLUI_STRU elements at the interface between the structure and the fluid. Moreover, an impedance condition is defined on the external border of the fluid, by a loading of type IMPE and the calculation of the impedance matrices that are linked to it.