Difference between revisions of "Porous absorber"

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Figure 1: Sketch of the geometry. The porous material is marked in blue, the acoustic source by

q

.

Figure 2: Frequency response function.

Description

The Porous absorber benchmark models the sound pressure in a cavity excited by a single harmonic load. One side of the cavity is covered by a layer of poroelastic material, which adds dissipation to the system. The geometry of this model follows ^[1]. Various projection-based model order reduction methods have been applied and compared using this example as a benchmark in ^[2].

The cavity has the dimensions $0.75 \times 0.6 \times 0.4\,\mathrm{m}$ and one wall is covered by a $0.05\,\mathrm{m}$ thick poroelastic layer acting as a sound absorber. The poroelastic material is described by the Biot theory^[3] and the system is excited by a point source located in a corner opposite of the porous layer. The material parameters for the acoustic fluid and the poroelastic material have been chosen according to^[1]. The transfer function measures the mean acoustic pressure inside the cavity.

Dimensions

System structure:

\begin{align} \left( K + \tilde{\gamma}(s) K_{p,1} + \tilde{\rho}_f(s) K_{p,2} + s^2 M + s^2 \tilde{\gamma}(s) M_{p,1} + s^2 \tilde{\rho}(s) M_{p,2} + \frac{s^2 \phi^2}{\tilde{R}(s)} M_{p,3} \right) x(s) &= B, \\ y(s) &= C x(s), \end{align}

with the frequency dependent functions for the effective densities $\tilde{\rho}(s), \tilde{\rho}_f(s)$ , the parameter $\tilde{\gamma}(s)$ relating the effective densities and the frequency dependent elasticity coefficients to the porosity, and the scaled effective bulk modulus $\tilde{R}(s)$ . For more details on the functions, see ^[1].

System dimensions:

$K, K_{p,1}, K_{p,2}, M, M_{p,1}, M_{p,2}, M_{p,3} \in \mathbb{R}^{n \times n}$ , $B \in \mathbb{R}^{n \times 1}$ , $C \in \mathbb{R}^{1 \times n}$ , with $n=386\,076$ .

Data

The data is available at Zenodo.

Remarks

The numerical model resembles the results from^[1] in a frequency range from $100\,\mathrm{Hz}$ to $1000\,\mathrm{Hz}$ . The frequency response in this range is also included in the dataset.
The finite element discretization has been performed with Kratos Multiphysics.
A comparison of different interpolation-based MOR methods using this benchmark example is available in^[2].

Citation

To cite this benchmark, use the following references:

For the benchmark itself and its data:

 @Misc{dataAum23,
   author =       {Aumann, Q.},
   title =        {Matrices for an acoustic cavity with poroelastic layer},
   howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
   year =         2023,
   doi =          {10.5281/zenodo.8087341}
 }

For the background on the benchmark:

 @Article{AumW23,
   author =       {Aumann, Q. and Werner, S.~W.~R.},
   title =        {Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods},
   journal =      {Journal of Sound and Vibration},
   volume =       543,
   year =         2023,
   pages =        {117363},
   doi =          {10.1016/j.jsv.2022.117363},
   publisher =    {Elsevier {BV}}
 }

References

↑ ^1.0 ^1.1 ^1.2 ^1.3 R. Rumpler, P. Göransson, J.-F. Deü. "A finite element approach combining a reduced-order system, Padé approximants, and an adaptive frequency windowing for fast multi-frequency solution of poro-acoustic problems", International Journal for Numerical Methods in Engineering, 97: 759-784, 2014.
↑ ^2.0 ^2.1 Q. Aumann, S. W. R. Werner. "Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods", Journal of Sound and Vibration, 543: 117363, 2023.
↑ M. A. Biot. "Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range", J. Acoust. Soc. Am., 28(2):168–178, 1956.

[rumpler14-1] 1.0 ^1.1 ^1.2 ^1.3 R. Rumpler, P. Göransson, J.-F. Deü. "A finite element approach combining a reduced-order system, Padé approximants, and an adaptive frequency windowing for fast multi-frequency solution of poro-acoustic problems", International Journal for Numerical Methods in Engineering, 97: 759-784, 2014.

[aumann23-2] 2.0 ^2.1 Q. Aumann, S. W. R. Werner. "Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods", Journal of Sound and Vibration, 543: 117363, 2023.

[biot56-3] M. A. Biot. "Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range", J. Acoust. Soc. Am., 28(2):168–178, 1956.

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
 {{preliminary}} <!-- Do not remove -->
+[[Category:benchmark]]
+[[Category:linear]]
+[[Category:SISO]]
+<figure id="fig:plot1">
+[[File:Porous_absorber.png|480px|thumb|right|<caption>Sketch of the geometry. The porous material is marked in blue, the acoustic source by <math>q</math>.</caption>]]
+</figure>
+<figure id="fig:tf">
+[[File:Porous_absorber_frf.png|480px|thumb|right|<caption>Frequency response function.</caption>]]
+</figure>
 ==Description==
 The '''Porous absorber''' benchmark models the sound pressure in a cavity excited by a single harmonic load. One side of the cavity is covered by a layer of poroelastic material, which adds dissipation to the system. The geometry of this model follows <ref name="rumpler14"/>. Various projection-based model order reduction methods have been applied and compared using this example as a benchmark in <ref name="aumann23"/>.
+The cavity has the dimensions <math>0.75 \times 0.6 \times 0.4\,\mathrm{m}</math> and one wall is covered by a <math>0.05\,\mathrm{m}</math> thick poroelastic layer acting as a sound absorber. The poroelastic material is described by the Biot theory<ref name="biot56"/> and the system is excited by a point source located in a corner opposite of the porous layer. The material parameters for the acoustic fluid and the poroelastic material have been chosen according to<ref name="rumpler14"/>. The transfer function measures the mean acoustic pressure inside the cavity.
+==Dimensions==
+System structure:
+:<math>
+\begin{align}
+\left( K + \tilde{\gamma}(s) K_{p,1} + \tilde{\rho}_f(s) K_{p,2} + s^2 M + s^2 \tilde{\gamma}(s) M_{p,1} + s^2 \tilde{\rho}(s) M_{p,2} + \frac{s^2 \phi^2}{\tilde{R}(s)} M_{p,3} \right) x(s) &= B, \\
+y(s) &= C x(s),
+\end{align}
+</math>
+with the frequency dependent functions for the effective densities <math>\tilde{\rho}(s), \tilde{\rho}_f(s)</math>, the parameter <math>\tilde{\gamma}(s)</math> relating the effective densities and the frequency dependent elasticity coefficients to the porosity, and the scaled effective bulk modulus <math>\tilde{R}(s)</math>. For more details on the functions, see <ref name="rumpler14"/>.
+System dimensions:
+<math>K, K_{p,1}, K_{p,2}, M, M_{p,1}, M_{p,2}, M_{p,3} \in \mathbb{R}^{n \times n}</math>,
+<math>B \in \mathbb{R}^{n \times 1}</math>,
+<math>C \in \mathbb{R}^{1 \times n}</math>,
+with <math>n=386\,076</math>.
+==Data==
+The data is available at [https://doi.org/10.5281/zenodo.8087341 Zenodo].
+==Remarks==
+* The numerical model resembles the results from<ref name="rumpler14"/> in a frequency range from <math>100\,\mathrm{Hz}</math> to <math>1000\,\mathrm{Hz}</math>. The frequency response in this range is also included in the dataset.
+* The finite element discretization has been performed with [https://github.com/KratosMultiphysics/Kratos Kratos Multiphysics].
+* A comparison of different interpolation-based MOR methods using this benchmark example is available in<ref name="aumann23"/>.
+==Citation==
+To cite this benchmark, use the following references:
+* For the benchmark itself and its data:
+  @Misc{dataAum23,
+    author =       {Aumann, Q.},
+    title =        {Matrices for an acoustic cavity with poroelastic layer},
+    howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
+    year =         2023,
+    doi =          {10.5281/zenodo.8087341}
+  }
+* For the background on the benchmark:
+  @Article{AumW23,
+    author =       {Aumann, Q. and Werner, S.~W.~R.},
+    title =        {Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods},
+    journal =      {Journal of Sound and Vibration},
+    volume =       543,
+    year =         2023,
+    pages =        {117363},
+    doi =          {10.1016/j.jsv.2022.117363},
+    publisher =    {Elsevier {BV}}
+  }
@@ Line 12: / Line 77: @@
 <ref name="aumann23">Q. Aumann, S. W. R. Werner. "<span class="plainlinks">[https://doi.org/10.1016/j.jsv.2022.117363 Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods]</span>", Journal of Sound and Vibration, 543: 117363, 2023.</ref>
+<ref name="biot56">M. A. Biot. "<span class="plainlinks">[http://dx.doi.org/10.1121/1.1908239 Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range]</span>", J. Acoust. Soc. Am., 28(2):168–178, 1956.</ref>
 </references>