1. Reference problem

1.1. Geometry

We consider a hollow sphere with an outer radius \({R}_{\mathit{ext}}=\mathrm{1,4}m\) and an inner radius \({R}_{\text{int}}=\mathrm{1,0}m\) 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 \({R}_{\mathit{bgt}}=\mathrm{3,0}m\) and an inner radius \({R}_{\mathit{ext}}=\mathrm{1,4}m\).

1.2. Material properties

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

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

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

1.3. Boundary conditions and loads

The stress considered in the problem is a strictly radial distributed pressure \({p}_{\mathit{imp}}=\mathrm{1,5}\mathit{MPa}\) imposed on the surface \({\mathrm{\Gamma }}_{\text{int}}\).

1.4. Initial conditions

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