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


Geometry
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We consider a hollow sphere with an outer radius :math:`{R}_{\mathit{ext}}=\mathrm{1,4}m` and an inner radius :math:`{R}_{\text{int}}=\mathrm{1,0}m` immersed in water. The fluid domain is considered to be infinite. But to model the problem, we truncate the fluid domain into a hollow sphere with an outer radius :math:`{R}_{\mathit{bgt}}=\mathrm{3,0}m` and an inner radius :math:`{R}_{\mathit{ext}}=\mathrm{1,4}m`.


Material properties
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The structure is isotropic elastic whose properties are: :math:`E=210000\mathit{MPa}`, :math:`\nu =0.3` and :math:`{\mathrm{\rho }}_{s}=7850\mathit{kg}\mathrm{.}{m}^{-3}`

The fluid is water whose density is :math:`{\mathrm{\rho }}_{f}=1000\mathit{kg}\mathrm{.}{m}^{-3}`. The speed of sound in this fluid is worth :math:`{c}_{f}=1500m\mathrm{.}{s}^{-1}`.

Condition BGT is imposed (IMPE_FACE) with the following acoustic impedance: :math:`{Z}_{c}=3\times {\mathrm{\rho }}_{f}+{\mathrm{\rho }}_{f}{c}_{f}i`. This quantity is, of course, complex.


Boundary conditions and loads
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The stress considered in the problem is a strictly radial distributed pressure :math:`{p}_{\mathit{imp}}=\mathrm{1,5}\mathit{MPa}` imposed on the surface :math:`{\mathrm{\Gamma }}_{\text{int}}`.


Initial conditions
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We start from a zero state (constraints, speeds, displacements and accelerations)