3. Modeling A

3.1. Characteristics of modeling

3.1.1. Equivalent vibro-acoustic system to be modelled

In order to avoid return waves coming from the ends of a modeling that is necessarily of a finite dimension, these ends are equipped with « piston-shock absorber » systems as in figure [Figure 3.1.1-a].

The channel is modelled over a total length of \(28m\) sufficient to obtain with certainty, at least the first two extrema of the piston displacement curve without disturbing a reflection wave at the ends.

Figure 3.1.1-a: Equivalent vibro-acoustic system

3.1.2. Numerical modeling in finite elements

We chose to model in 2D.

For the fluid: the modeling is in formulation \(p,\phi\).

It is achieved by assigning elements PHENOMENE =” MECANIQUE “, MODELISATION =”2D_ FLUIDE” to QUAD4 cells (quadrilaterals with 4 nodes).

For structures: the modeling is in formulation \(u\).

It is achieved by assigning elements PHENOMENE =” MECANIQUE “, MODELISATION =”D_ PLAN” to QUAD4 cells (quadrilaterals with 4 nodes).

For discrete oscillator elements: the modeling is in formulation \(u\).

It is carried out by assigning elements PHENOMENE =” MECANIQUE “, MODELISATION =” DIS_T” to point POI1 cells.

For fluid-structure interfaces: the modeling is in formulation \(u,p,\phi\).

It is achieved by assigning elements PHENOMENE =” MECANIQUE “, MODELISATION =”2D_ FLUIDE_STRU” to SEG2 meshes (segments with 2 nodes).

3.2. Characteristics of the mesh

Figure 3.2-a: Two-dimensional mesh of the fluidelastic oscillator model

The data characterizing this modeling have been grouped together in the table below.

	Type of mesh			Number
	QUAD4	SEG2	POI1	total
Number of elements	2870	80	3	2953
Number of nodes generated	3164	0	0	3164

Table 3.2-1: Characteristics of the two-dimensional mesh of the fluidelastic oscillator

3.3. Calculus

It is desired to validate fluid-structure interaction elements in transient conditions by an excitation loading. The displacement of the wall piston is calculated with the operator DYNA_VIBRA. A pressure of \(17\mathit{MPa}\) is applied to the piston.

Note: to validate the loads on the fluid-structure part, in this modeling the pressure is applied to the interface and not to the structure. Since the structure is supposed to be infinitely rigid, it is strictly equivalent.

3.4. Tested sizes and results

The results of the calculation with code_aster are presented graphically on the [Figure 4.1-a] superimposed with the « analytical » reference solution.

The code_aster curve seems very close to the reference during the first four oscillations but the differences, both in amplitude and in phase, are more and more noticeable as \(t\) increases.

Figure 4.1-a: Comparison between calculation Code_Aster and semi-analytical reference

The test relates to the movement of the wall piston in two given moments close to the first two extremes.

The table shows a comparison of the first two extremes of the piston displacement curve between analytical points and points calculated by code_aster.

The values obtained from the extreme moments in both cases are estimated values extracted without interpolation of the raw calculated values: they do not correspond exactly between the analytical curve and the code_aster curve.

The tolerance for relative deviation from the analytical value is estimated at 1.%.

	Reference analytic
	Inst. (\(\mathrm{ms}\))	Depl. (\(\mathrm{mm}\))
1st Extremum	20,13	—1.3530
2MEExtremum	26.05	—0.4210

Code non-regression test:

the tolerance for relative deviation from the reference is equal to 0.1%.

3.5. note

The reference values finally retained are those obtained by code_aster when restoring the test case, which will therefore make it possible to verify the subsequent non-regression of the code during its evolution.