Benchmark solution
=====================


Benchmark results
----------------------

The frequency method for coupling between Miss3D and *Code_Aster* is described in the reference document [:ref:`bib1 <bib1>`].

We test the modulus of the radial displacements obtained outside and inside the ball with respect to an analytical solution calculated and detailed in an application study [:ref:`bib2 <bib2>`]. Solutions under pressure and development only depend on radius and time. It is considered that the total pressure in the fluid is due to the sum of two contributions:


* pressure :math:`{p}_{d}` due to the vibration of the wall at the interface with the solid medium,


* pressure :math:`{p}_{0}` due to the action of the Dirac mass at the center of the sphere, in an infinite fluid


* :math:`p={p}_{d}+{p}_{0}`

The Helmholtz equation for the propagation of waves in the fluid in the absence of a source is written in spherical coordinates after Fourier transformation:

:math:`\frac{1}{{r}^{2}}\frac{\delta }{\delta r}({r}^{2}\frac{\delta {p}_{d}}{\delta r})(r,\omega )+\frac{\omega }{{c}_{f}^{2}}{p}_{d}(r,\omega )=0`


    * By asking: :math:`{k}_{f}(\omega )=\frac{\omega }{{c}_{f}}`, we get a solution of the form :math:`{p}_{d}=A(\mathrm{sin}\frac{({k}_{f}r)}{4\pi r})`,

The solution for pressure :math:`{p}_{0}` is given by a Green function and the pressure in the fluid is written as: :math:`p=\frac{{e}^{i{k}_{f}r}}{4\pi r}+A(\mathrm{sin}\frac{({k}_{f}r)}{4\pi r})`

The Navier equation for the propagation of waves in the solid in the absence of a source is written in spherical coordinates after Fourier transformation and by changing variables :math:`u=\frac{\delta \phi }{\delta r}`:

:math:`\frac{{\delta }^{2}}{\delta {r}^{2}}(r\phi )(r,\omega )+{k}_{p}^{2}(r\phi )(r,\omega )=0`


    * By setting :math:`{k}_{p}(\omega )=\frac{\omega }{{c}_{p}}`, we get a solution of the form:

:math:`u=B\frac{{e}^{i{k}_{p}r}}{4\pi r}(\frac{i{k}_{p}r-1}{r})+C\frac{{e}^{\text{-}i{k}_{p}r}}{4\pi r}(\frac{-i{k}_{p}r-1}{r})`.

The 3 unknown coefficients :math:`A`, :math:`B` and :math:`C` are then determined on the basis of 3 limit conditions:


* Continuity of normal movements at the ground/fluid interface :math:`\rho {\omega }^{2}u=\mathrm{grad}(p)` for :math:`r={r}_{1}`,


* Continuity of normal stresses at the soil-fluid interface :math:`{\sigma }_{\mathrm{rr}}+p=0` for :math:`r={r}_{1}`,


* Zero radial stress on the external surface zero :math:`{\sigma }_{\mathrm{rr}}=0` for :math:`r={r}_{2}`,


Bibliographical references
---------------------------


1. D. CLOUTEAU: "MISS3D reference manual — version 6.3 — Centrale Recherche SA"


2. G. DEVESA, M. FESTA: "study with the*Code_Aster* and its interface with MISS3D of Soil-Structure-Fluid interaction: Application to the dynamic calculation of arch dams", EDF /R &D HP-52/99/001/A.