2. Benchmark solution#

The solution should show the absorption of a compression wave by the absorbent surface. The imposed displacement is a uniform translation along the \(x\) axis. We must obtain an identical field of movement in this direction in all \(x=\mathrm{Cte}\) planes. Moreover, the absorbent border is orthogonal to this axis. The absorption of plane compression waves under normal incidence is therefore studied. The [bib1] theory says that with a solid paraxial boundary of order 0, this absorption is perfect. This is what we need to check with this reference solution.

By observing the evolution of the displacement at a given point in the mesh, we will therefore try to find in the signal obtained the duration of excitation and the return to rest after the passage of the wave, characteristic of its absorption.

2.1. Benchmark results#

In this paragraph we give the results obtained with*Code_Aster* in this configuration. We check that they are satisfactory and we take them as a reference for the future.

For the 3D case, they concern the bar having a length of \(200m\), the evolution of the movement in \(x\) at a point on the bar located \(150m\) from the excited face in the direction \(x\) and at the center of the section in the plane \(\mathrm{yz}\). For the 2D case, the bar having a length of \(50m\), the point is located \(40m\) from the face according to \(x\) and in the middle of the section in the direction

_images/Object_8.svg

(in 2D, we use a shorter and more refined mesh).

_images/10000000000003F0000002609959046F198A8EB2.png _images/100000000000041B00000236676A47DADAB1F87E.png

As expected, the width of the signal measured in both cases is identical to that of the excitation function. Physically, the propagation of the compression wave is well observed. The signal is slightly modified in its propagation and the maximum amplitude of 1 mm is therefore recovered. We also clearly note the return to rest immediately after the wave has passed and the absence of a signal reflected at the end of the mesh.

2.2. Uncertainties#

It is a numerical study result. We find the qualitative forecasts. Numeric values are related to the accuracy of the calculation. Only the return to rest is precisely given by the analysis.

2.3. Bibliographical references#

    1. MODARESSI « Numerical modeling of wave propagation in elastic porous media. » Doctor-engineer thesis, Ecole Centrale de Paris (1987).