2. Reference problem#

The aim is to determine the monodirectional response of an elastic bar to a compression wave in order to assess the effectiveness of method CALM in the absorption of an elastic wave on a domain boundary. We therefore seek to evaluate the ability of method CALM to model radiative damping. For this, we also compare ourselves to the absorption capacity of absorbent borders.

2.1. Geometry#

We consider an elastic bar oriented along the global axis \(X\). In v2.03.201-Geom1 we show the geometry of the bar:

  • \(L_1\) is the length of the bar

  • \(L_2\) is the length of layer CALM

  • \(H\) is the height of the rung

  • \(A\) is the post-processing node, which is positioned in the middle of the bar

In tab_geometrie the geometric measurements of the bar are provided.

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In addition to the geometric configuration shown in v2.03.201-Geom1, a second configuration presented in v2.03.201-Geom2 is studied.

It is known that absorbent borders progressively decrease their capacity to absorb elastic waves as soon as the incidence of the wave is no longer orthogonal to the border. In particular, as soon as an angle of incidence of \(30°\) is exceeded, the absorption capacity decreases drastically. This configuration is useful for us to assess the effectiveness of the CALM method in relation to absorbent borders. The geometric configuration in question has an angle \(\phi=70°\).

2.2. Material properties#

The bar is composed of an elastic material with characteristics that are summarized in tab_caracte, where:

  • \(E\) is Young’s modulus

  • \(\rho\) is volume density

  • \(\nu\) is Poisson’s ratio

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2.3. Modeling and boundary conditions#

The elastic bar is modelled in plane deformations (D_ PLAN) with a linear mesh composed of QUA4 square elements of size \(l_{ef}=1\) m. To simulate monodirectional propagation, all degrees of freedom in direction \(Y\) are imposed that are zero.

2.4. Loading#

The transitory signal is imposed in two different ways:

  • For modeling A, B, C and D we impose a \(d_{impo}\) displacement on the edge \(BC\) (v2.03.201-Geom1)

  • For E and F modeling, a plane wave is imposed in the direction \(-X\) on the DE border (v2.03.201-Geom1). We recall that the plane wave is imposed in code_aster via an absorbing border and in particular via a time speed function. The signal used in the A, B, C and D models is therefore integrated.

The imposed load is a Ricker wavelet, which can be written in the following form:

\[R (t) =( 1-2 (\ pi f (t-t_0))) ^2) e^ {- (\ pi f (t-t_0))) ^2} :label: eq:ricker\]

where:

  • \(t_0=0.6\) s

  • \(f=3\) Hz

The time step considered is \(\Delta t=0.002\) s. In v2.03.201-Ricker we show the Ricker wavelet used in this modeling.

2.5. The modeling of layer CALM#

The main idea of method CALM is to model a damping that increases progressively throughout the layer to absorb the waves and without generating too high a contrast that could generate parasitic reflections. To do this, in code_aster we can define material characteristics that are a function of geometry. Depreciation is therefore defined in two steps:

  • The first step is to define a constant Rayleigh damping in the elastic bar. To do this, we set a \(\xi_0=0.2\%\) damping between \(1\) and \(10\) Hz

  • Then, in layer CALM, we change a Rayleigh damping which is equal to \(\xi_0\%\) for \(x=L_1\) and which is equal to \(\xi_{max}=400\%\) for \(x=L_1+L_2\) with a parabolic progression dependent on the coordinate \(x\)

Note

In the geometric configuration of the bar with an inclined border (v2.03.201-Geom2) we will also have a dependence on the coordinate \(y\).