2. Benchmark solution#
2.1. Benchmark results#
The frequency method for coupling between Miss3D and Code_Aster is described in the reference document [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 [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 \({p}_{d}\) due to the vibration of the wall at the interface with the solid medium,
pressure \({p}_{0}\) due to the action of the Dirac mass at the center of the sphere, in an infinite fluid
\(p={p}_{d}+{p}_{0}\)
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:
\(\frac{1}{{r}^{2}}\frac{\delta }{\delta r}({r}^{2}\frac{\delta {p}_{d}}{\delta r})(r,\omega )+\frac{\omega }{{c}_{f}^{2}}{p}_{d}(r,\omega )=0\)
By asking: \({k}_{f}(\omega )=\frac{\omega }{{c}_{f}}\), we get a solution of the form \({p}_{d}=A(\mathrm{sin}\frac{({k}_{f}r)}{4\pi r})\),
The solution for pressure \({p}_{0}\) is given by a Green function and the pressure in the fluid is written as: \(p=\frac{{e}^{i{k}_{f}r}}{4\pi r}+A(\mathrm{sin}\frac{({k}_{f}r)}{4\pi r})\)
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 \(u=\frac{\delta \phi }{\delta r}\):
\(\frac{{\delta }^{2}}{\delta {r}^{2}}(r\phi )(r,\omega )+{k}_{p}^{2}(r\phi )(r,\omega )=0\)
By setting \({k}_{p}(\omega )=\frac{\omega }{{c}_{p}}\), we get a solution of the form:
\(u=B\frac{{e}^{i{k}_{p}r}}{4\pi r}(\frac{i{k}_{p}r-1}{r})+C\frac{{e}^{\text{-}i{k}_{p}r}}{4\pi r}(\frac{-i{k}_{p}r-1}{r})\).
The 3 unknown coefficients \(A\), \(B\) and \(C\) are then determined on the basis of 3 limit conditions:
Continuity of normal movements at the ground/fluid interface \(\rho {\omega }^{2}u=\mathrm{grad}(p)\) for \(r={r}_{1}\),
Continuity of normal stresses at the soil-fluid interface \({\sigma }_{\mathrm{rr}}+p=0\) for \(r={r}_{1}\),
Zero radial stress on the external surface zero \({\sigma }_{\mathrm{rr}}=0\) for \(r={r}_{2}\),
2.2. Bibliographical references#
CLOUTEAU: « MISS3D reference manual — version 6.3 — Centrale Recherche SA »
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.