# Homogenization of the vibro-acoustic transmission on perforated plates with embedded resonators¶

## Mathematical model¶

This example presents the implementation of the homogenized model of the acoustic transmission on perforated plates described in [RohanLukes2021]. The elastic plate interacting with an inviscid fluid is replaced by an interface on which the homogenized transmission conditions are applied, see Fig. 1.

The frequency dependent homogenized coefficients are computed using characteristic responses of the representative cell which consists of the fluid part and the solid plate embedding the elastic inclusion and the resonator as illustrated in Fig. 2.

Equations describing the homogenized transmission layer involve the homogenized coefficients and are solved in the macroscopic domain and are coupled with the global acoustic field defined in .

## Numerical implementation¶

The problem of homogenized coefficients and the global macroscopic equations
are discretized and solved by means of the finite element method. The frequency
dependent coefficients are defined in `acoustics_micro.py`

and the
frequency independent coefficients calculated within the 2D plate
representation are specified in `acoustics_micro_plate.py`

.
The homogenization engine of SfePy, see [CimrmanLukesRohan2019], is employed
for their effective calculation.

At the macroscopic level, we consider the waveguide which domain is divided by
the perforated plate into two parts of the same shape and size, see
Fig. 3. The waveguide input is labelled by and the
incident wave is imposed on this boundary. The anechoic boundary condition is
considered at the waveguide output which is labelled by .
The definition of the macroscopic problem is in files
`acoustics_macro.py`

and `acoustics_macro_plate.py`

. The first file
specifies the equations and variables related to ,
while the second one defines the equations associated with the
homogenized interface .

The whole two-scale analysis is govern by the `acoustics.py`

script which
invokes the homogenization procedures and runs the simulation at the
macroscopic level.

## Running simulation¶

To run the numerical simulation, download the archive, unpack it in the main SfePy directory and type:

```
python example_vibroacoustics-2/acoustics.py
```

By running the `resview.py`

script, you can visualize the distribution of
the global pressure field calculated for frequency Hz:

```
./resview.py example_vibroacoustics-2/results/waveguide_mesh_w33000_p.vtk -v "270,90"
```

and e.g. the deflection field calculated at the interface :

```
./resview.py example_vibroacoustics-2/results/waveguide_mesh_w33000_dp0.vtk -v "0,0" --position-vector "0,2,0" -f real.w:p0 imag.w:p1
```

## References¶

[RohanLukes2021] | Rohan E., Lukeš V. Homogenization of the vibro-acoustic transmission on periodically perforated elastic plates with arrays of resonators. Applied Mathematical Modelling, 2022, DOI:10.1016/j.apm.2022.05.040 |

[CimrmanLukesRohan2019] | Cimrman R., Lukeš V., Rohan E. Multiscale finite element calculations in Python using SfePy. Advances in Computational Mathematics, 45(4):1897-1921, 2019, DOI:10.1007/s10444-019-09666-0 |