Porous absorber: Difference between revisions

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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 $0.75 \times 0.6 \times 0.4 m$ and one wall is covered by a $0.05 m$ 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].

Dimensions

System structure:

\begin{aligned} (A_{1} + \tilde{γ} (s) A_{2} + {\tilde{ρ}}_{f} (s) A_{3} + s^{2} A_{4} + s^{2} \tilde{γ} (s) A_{5} + s^{2} \tilde{ρ} (s) A_{6} + \frac{s^{2} ϕ^{2}}{\tilde{R} (s)} A_{7}) x (s) & = B, \\ y (s) & = C x (s), \end{aligned}

with the frequency dependent functions for the effective densities $\tilde{ρ} (s), {\tilde{ρ}}_{f} (s)$ , the parameter $\tilde{γ} (s)$ 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:

$A_{i}, \in ℝ^{n \times n}$ , with $i = 1, \dots, 7$ , $B \in ℝ^{n \times 1}$ , $C \in ℝ^{1 \times n}$ , with $n = 386 076$ .

Data

The data is available at Zenodo.

References

↑ ^1.0 ^1.1 ^1.2 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.
↑ 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 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] 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.

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[2]

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@@ Line 12: / Line 12: @@
 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 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"/>.
+==Dimensions==
+System structure:
+:<math>
+\begin{align}
+\left( A_1 + \tilde{\gamma}(s) A_2 + \tilde{\rho}_f(s) A_3 + s^2 A_4 + s^2 \tilde{\gamma}(s) A_5 + s^2 \tilde{\rho}(s) A_6 + \frac{s^2 \phi^2}{\tilde{R}(s)} A_7 \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>A_i, \in \mathbb{R}^{n \times n}</math>, with <math>i=1, \ldots, 7 </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].
 ==References==