MOR Wiki - User contributions [en]

Machine tool with relative movements

2023-07-13T13:30:25Z

Aumann:

{{preliminary}} 
[[Category:benchmark]]
[[Category:Sparse]]
[[Category:first differential order]]
[[Category:MIMO]]
[[Category:CRC-TR-96]]

<figure id="fig:plot1">
[[File:mt_geo.png|480px|thumb|right|<caption>Sketch of the geometry and the boundary conditions.</caption>]]
</figure>
==Description==

This benchmark models a machine tool composed out of two components. The "rail" part is the machine bed with a guide rail, the "slider" part moves along the rail on four carts and carries the tool.
Slider and rail move relative to each other during the simulated work process. This relative movement is realized by decomposing a weighting function into a Fourier series and superposing their weighted sum, which depends on the current position of the slider. The weighted sums are then used in every time step to compute the thermal load exchanged between the two components. This approach is described in more detail in<ref name="hernandez"/>. Due this approach, additional inputs and outputs are created, but the resulting system is still LTI, despite the relative movement.

<figure id="fig:plot2">
[[File:mt_rail.png|480px|thumb|right|<caption>Temperature evaluation on the rail midpoint.</caption>]]
</figure>
<figure id="fig:plot3">
[[File:mt_slider.png|480px|thumb|right|<caption>Temperature evaluation on the TCP.</caption>]]
</figure>
The thermal response of the system has been investigated in<ref name="ictimt"/>. Here, the temperature change in the middle of the rail and the tool center point (TCP), respectively, have been measured during a simulated work process. In the first <math>600\,\mathrm{s}</math>, the slider is moving from left to right constantly. Afterwards, the slider remains at its location. During the first <math>600\,\mathrm{s}</math> six heat sources introduce energy into the system. They are located at the TCP, the work piece, and the four carts (see Fig. 1). After the slider stops to move, the heat sources are switched off. The temperature evaluations are given in Figs. 2 and 3.

==Dimensions==
Both systems have the same structure:

:<math>
\begin{align}
E \dot{x}(t) &= A x(t) + B u(t), \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:
<math>E \in \mathbb{R}^{n \times n}</math>,
<math>A \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{p \times n}</math>,
with <math>n=91\,181</math>, <math>m=p=89</math> for the rail system and <math>n=74\,392</math>, <math>m=p=8</math> for the slider system.

==Data==
The data is available at [https://doi.org/10.5281/zenodo.8099978 Zenodo].

==Remarks==
* The dataset contains also the data for the movement profile that was used to produce the results reported here. This can be changed by modifying the variable <code>sc</code>.
* The system also contains the elastic degrees of freedom and their coupling in <math>x</math>-direction.

==Origin==
The '''Machine tool with relative movements''' model was developed in the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Category:CRC-TR-96 CRC/TR 96] Project-ID 174223256 financed by the German Research Foundation DFG.

==Citation==
To cite this benchmark, use the following references:

* For the benchmark itself and its data:
@misc{dataAumSV23,
author = {Q. Aumann and J. Saak and J. Vettermann},
title = {Model of a machine tool with relative movements},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2023,
doi = {10.5281/zenodo.8099979}
}

* For the background on the benchmark:

@article{AumBSetal23,
author = {Q. Aumann and P. Benner and J. Saak and J. Vettermann},
booktitle = {Lecture Notes in Production Engineering},
publisher = {Springer International Publishing},
title = {Model order reduction strategies for the computation of compact machine tool models},
year = {2023},
editor = {S. Ihlenfeldt},
pages = {132--145},
doi = {10.1007/978-3-031-34486-2_10},
}

==References==

<references>

<ref name="ictimt">Q. Aumann, P. Benner, J. Saak, J. Vettermann. "[https://doi.org/10.1007/978-3-031-34486-2_10 Model Order Reduction Strategies for the Computation of Compact Machine Tool Models]", In: S. Ihlenfeldt (ed) Lecture Notes in Production Engineering. Springer International Publishing, pp 132–145, 2023.</ref>

<ref name="hernandez">P. Hernández-Becerro [https://www.research-collection.ethz.ch/handle/20.500.11850/449279 Efficient Thermal Error Models of Machine Tools"], Dissertation, ETH Zurich, 2002.</ref>

</references>

Machine tool with relative movements

2023-06-30T12:39:18Z

Aumann:

{{preliminary}} 
[[Category:benchmark]]
[[Category:Sparse]]
[[Category:first differential order]]
[[Category:MIMO]]
[[Category:CRC-TR-96]]

<figure id="fig:plot1">
[[File:mt_geo.png|480px|thumb|right|<caption>Sketch of the geometry and the boundary conditions.</caption>]]
</figure>
==Description==

This benchmark models a machine tool composed out of two components. The "rail" part is the machine bed with a guide rail, the "slider" part moves along the rail on four carts and carries the tool.
Slider and rail move relative to each other during the simulated work process. This relative movement is realized by decomposing a weighting function into a Fourier series and superposing their weighted sum, which depends on the current position of the slider. The weighted sums are then used in every time step to compute the thermal load exchanged between the two components. This approach is described in more detail in<ref name="hernandez"/>. Due this approach, additional inputs and outputs are created, but the resulting system is still LTI, despite the relative movement.

<figure id="fig:plot2">
[[File:mt_rail.png|480px|thumb|right|<caption>Temperature evaluation on the rail midpoint.</caption>]]
</figure>
<figure id="fig:plot3">
[[File:mt_slider.png|480px|thumb|right|<caption>Temperature evaluation on the TCP.</caption>]]
</figure>
The thermal response of the system has been investigated in<ref name="ictimt"/>. Here, the temperature change in the middle of the rail and the tool center point (TCP), respectively, have been measured during a simulated work process. In the first <math>600\,\mathrm{s}</math>, the slider is moving from left to right constantly. Afterwards, the slider remains at its location. During the first <math>600\,\mathrm{s}</math> six heat sources introduce energy into the system. They are located at the TCP, the work piece, and the four carts (see Fig. 1). After the slider stops to move, the heat sources are switched off. The temperature evaluations are given in Figs. 2 and 3.

==Dimensions==
Both systems have the same structure:

:<math>
\begin{align}
E \dot{x}(t) &= A x(t) + B u(t), \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:
<math>E \in \mathbb{R}^{n \times n}</math>,
<math>A \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{p \times n}</math>,
with <math>n=91\,181</math>, <math>m=p=89</math> for the rail system and <math>n=74\,392</math>, <math>m=p=8</math> for the slider system.

==Data==
The data is available at [https://doi.org/10.5281/zenodo.7300346 Zenodo].

==Remarks==
* The dataset contains also the data for the movement profile that was used to produce the results reported here. This can be changed by modifying the variable <code>sc</code>.
* The system also contains the elastic degrees of freedom and their coupling in <math>x</math>-direction.

==Origin==
The '''Machine tool with relative movements''' model was developed in the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Category:CRC-TR-96 CRC/TR 96] Project-ID 174223256 financed by the German Research Foundation DFG.

==Citation==
To cite this benchmark, use the following references:

* For the benchmark itself and its data:
@misc{dataAumSV23,
author = {Q. Aumann and J. Saak and J. Vettermann},
title = {Model of a machine tool with relative movements},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2023,
doi = {10.5281/zenodo.8099979}
}

* For the background on the benchmark:

@article{AumBSetal23,
author = {Q. Aumann and P. Benner and J. Saak and J. Vettermann},
booktitle = {Lecture Notes in Production Engineering},
publisher = {Springer International Publishing},
title = {Model order reduction strategies for the computation of compact machine tool models},
year = {2023},
editor = {S. Ihlenfeldt},
pages = {132--145},
doi = {10.1007/978-3-031-34486-2_10},
}

==References==

<references>

<ref name="ictimt">Q. Aumann, P. Benner, J. Saak, J. Vettermann. "[https://doi.org/10.1007/978-3-031-34486-2_10 Model Order Reduction Strategies for the Computation of Compact Machine Tool Models]", In: S. Ihlenfeldt (ed) Lecture Notes in Production Engineering. Springer International Publishing, pp 132–145, 2023.</ref>

<ref name="hernandez">P. Hernández-Becerro [https://www.research-collection.ethz.ch/handle/20.500.11850/449279 Efficient Thermal Error Models of Machine Tools"], Dissertation, ETH Zurich, 2002.</ref>

</references>

File:Mt slider.png

2023-06-30T12:18:18Z

Aumann:

File:Mt rail.png

2023-06-30T12:18:01Z

Aumann:

File:Mt geo.png

2023-06-30T12:08:44Z

Aumann:

Machine tool with relative movements

2023-06-30T12:08:15Z

Aumann: Created page with "{{preliminary}}  ==Description== This benchmark models a machine tool composed out of two components. The "rail" part is the machine bed with a guide ra..."

{{preliminary}} 

==Description==
This benchmark models a machine tool composed out of two components. The "rail" part is the machine bed with a guide rail, the "slider" part moves along the rail on four carts and carries the tool.
Slider and rail move relative to each other during the simulated work process. This relative movement is realized by decomposing a weighting function into a Fourier series and superposing their weighted sum, which depends on the current position of the slider. The weighted sums are then used in every time step to compute the thermal load exchanged between the two components. This approach is described in more detail in<ref=""/>. The thermal response of the system has been investigated in<ref=""/>.

<figure id="fig:plot1">
[[File:mt_geo.png|480px|thumb|right|<caption>Sketch of the geometry and the boundary conditions.</caption>]]
</figure>

The relative movement between rail and slider is modeled via a superposition of inputs and outputs. The measurement outputs have indices i=45 for the rail and i=4 for the slider and measure the temperature change in the middle of the rail and the tool center point (TCP), respectively.

Porous absorber

2023-06-29T08:28:56Z

Aumann:

{{preliminary}} 
[[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}}
}

==References==

<references>

<ref name="rumpler14"> R. Rumpler, P. Göransson, J.-F. Deü. "[http://dx.doi.org/10.1002/nme.4609 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.</ref>

<ref name="aumann23">Q. Aumann, S. W. R. Werner. "[https://doi.org/10.1016/j.jsv.2022.117363 Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods]", Journal of Sound and Vibration, 543: 117363, 2023.</ref>

<ref name="biot56">M. A. Biot. "[http://dx.doi.org/10.1121/1.1908239 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.</ref>

</references>

Porous absorber

2023-06-29T08:25:23Z

Aumann:

{{preliminary}} 
[[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].

==References==

<references>

<ref name="rumpler14"> R. Rumpler, P. Göransson, J.-F. Deü. "[http://dx.doi.org/10.1002/nme.4609 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.</ref>

<ref name="aumann23">Q. Aumann, S. W. R. Werner. "[https://doi.org/10.1016/j.jsv.2022.117363 Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods]", Journal of Sound and Vibration, 543: 117363, 2023.</ref>

<ref name="biot56">M. A. Biot. "[http://dx.doi.org/10.1121/1.1908239 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.</ref>

</references>

Porous absorber

2023-06-27T15:25:29Z

Aumann:

{{preliminary}} 

<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"/>.

==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].

==References==

<references>

<ref name="rumpler14"> R. Rumpler, P. Göransson, J.-F. Deü. "[http://dx.doi.org/10.1002/nme.4609 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.</ref>

<ref name="aumann23">Q. Aumann, S. W. R. Werner. "[https://doi.org/10.1016/j.jsv.2022.117363 Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods]", Journal of Sound and Vibration, 543: 117363, 2023.</ref>

<ref name="biot56">M. A. Biot. "[http://dx.doi.org/10.1121/1.1908239 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.</ref>

</references>

Porous absorber

2023-06-27T15:17:55Z

Aumann:

{{preliminary}} 

<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"/>.

==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==

<references>

<ref name="rumpler14"> R. Rumpler, P. Göransson, J.-F. Deü. "[http://dx.doi.org/10.1002/nme.4609 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.</ref>

<ref name="aumann23">Q. Aumann, S. W. R. Werner. "[https://doi.org/10.1016/j.jsv.2022.117363 Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods]", Journal of Sound and Vibration, 543: 117363, 2023.</ref>

<ref name="biot56">M. A. Biot. "[http://dx.doi.org/10.1121/1.1908239 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.</ref>

</references>

Porous absorber

2023-06-27T14:09:14Z

Aumann:

File:Porous absorber frf.png

2023-06-27T14:02:49Z

Aumann:

File:Porous absorber.png

2023-06-27T14:01:52Z

Aumann:

Porous absorber

2023-06-27T14:01:33Z

Aumann:

Porous absorber

2023-05-23T07:41:09Z

Aumann: Created page with "{{preliminary}}  ==Description== The '''Porous absorber''' benchmark models the sound pressure in a cavity excited by a single harmonic load. One side o..."

{{preliminary}} 

==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"/>.

==References==

<references>

<ref name="rumpler14"> R. Rumpler, P. Göransson, J.-F. Deü. "[http://dx.doi.org/10.1002/nme.4609 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.</ref>

<ref name="aumann23">Q. Aumann, S. W. R. Werner. "[https://doi.org/10.1016/j.jsv.2022.117363 Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods]", Journal of Sound and Vibration, 543: 117363, 2023.</ref>

</references>

Plate with tuned vibration absorbers

2023-02-23T18:39:12Z

Aumann: Update dataset doi

{{preliminary}} 
[[Category:benchmark]]
[[Category:linear]]
[[Category:second differential order]]
[[Category:SISO]]
[[Category:MIMO]]

<figure id="fig:plot1">
[[File:Plate_tva.png|480px|thumb|right|<caption>Sketch of the geometry.</caption>]]
</figure>

<figure id="fig:tf">
[[File:Plate_tva_frf.png|480px|thumb|right|<caption>Frequency response function.</caption>]]
</figure>

==Description==
The '''Plate with tuned vibration absorbers''' benchmark models the vibration response of a plate excited by a single harmonic load. The plate is equipped with 108 tuned vibration absorbers (TVA), which change the vibration pattern of the host structure in a narrow frequency band around their tuning frequency<ref name="sun95"/>. Such systems have, for example, been examined in <ref name="jagodzinski20"/> and <ref name="claeys16"/>.

This benchmark models an aluminum plate with dimensions <math>0.8 \times 0.8\,\mathrm{m}</math> and a thickness of <math>t = 1\,\mathrm{mm}</math>. The surrounding edges are simply supported. The plate's surface is equipped with six struts along the <math>y</math>-direction, on which the TVAs are placed. The TVAs have a combined mass of <math>10\,\%</math> of the plate mass and are tuned to <math>f=48\,\mathrm{Hz}</math>. The plate is excited by a single load near one of the corners of the plate (see sketch). The root mean square of the displacement in <math>z</math>-direction at all points of the plate surface is plotted in Figure 2. The effect of the TVAs is clearly visible in the frequency range around their tuning frequency.

The following material parameters have been considered for aluminum:
{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Parameter
|Value
|Unit
|-style="background-color:#FFFFFF;"
|<math>E</math>
|<math>69</math>
|<math>\mathrm{GPa}</math>
|-style="background-color:#FFFFFF;"
|<math>\rho</math>
|<math>2650</math>
|<math>\mathrm{kg}\,\mathrm{m}^{-3}</math>
|-style="background-color:#FFFFFF;"
|<math>\nu</math>
|<math>0.22</math>
|<math>-</math>
|-
|}

==Dimensions==
System structure:

:<math>
\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>M \in \mathbb{R}^{n \times n}</math>,
<math>E \in \mathbb{R}^{n \times n}</math>,
<math>K \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times 1}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
with <math>n=201\,900</math> and <math>q=28\,278</math>.

Proportional damping, i.e. <math>E=\alpha M + \beta K</math>, with <math>\alpha=1\cdot 10^{-2}, \beta=1\cdot 10^{-4}</math> is considered.
The matrices <math>M, E, K</math> are positive (semi-) definite.

==Data==
The data is available at [https://doi.org/10.5281/zenodo.7507011 Zenodo].

==Remarks==
* The dataset also contains a version with a single output (SISO). Here, the displacement of the plate at the location of the load is evaluated.
* The frequency response in the range <math>1\,\mathrm{Hz}</math> to <math>250\,\mathrm{Hz}</math> is 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 a plate with tuned vibration absorbers},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2023,
doi = {10.5281/zenodo.7507011}
}

* 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==

<references>

<ref name="sun95"> J. Q. Sun, M. R. Jolly, M. A. Norris. "[http://dx.doi.org/10.1115/1.2836462 Passive, Adaptive and Active Tuned Vibration Absorbers—A Survey]", Journal of Mechanical Design, 117.B: 234–242, 1995.</ref>

<ref name="jagodzinski20"> D. J. Jagodzinski, M. Miksch, Q. Aumann, G. Müller. "[http://dx.doi.org/10.1080/15397734.2020.1787842 Modeling and optimizing an acoustic metamaterial to minimize low-frequency structure-borne sound]", Mechanics Based Design of Structures and Machines, 50(8): 2877–2891, 2020.</ref>

<ref name="claeys16"> C. Claeys, E. Deckers, B. Pluymers, W. Desmet. "[http://dx.doi.org/10.1016/j.ymssp.2015.08.029 A lightweight vibro-acoustic metamaterial demonstrator: Numerical and experimental investigation]", Mechanical Systems and Signal Processing, 70-71: 853–880, 2016.</ref>

<ref name="aumann23">Q. Aumann, S. W. R. Werner. "[https://doi.org/10.1016/j.jsv.2022.117363 Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods]", Journal of Sound and Vibration, 543: 117363, 2023.</ref>

</references>

Sound transmission through a plate

2023-02-23T13:47:29Z

Aumann: Update doi

{{preliminary}} 
[[Category:benchmark]]
[[Category:linear]]
[[Category:second differential order]]
[[Category:SISO]]

<figure id="fig:plot1">
[[File:guy.png|480px|thumb|right|<caption>Sketch of the geometry.</caption>]]
</figure>

<figure id="fig:tf">
[[File:guy_tf.png|480px|thumb|right|<caption>Transfer function.</caption>]]
</figure>

==Description==
The '''Sound transmission through a plate''' benchmark models the radiation of a vibrating plate and the excitation of a structure by an oscillating acoustic fluid. It is based on an experiment by Guy<ref name="guy81"/>.

The system consists of a cuboid acoustic cavity, where one wall is considered a system of two parallel elastic brass plates with a <math>2\,\mathrm{cm}</math> air gap between them; all other walls are considered rigid. The plates measure <math>0.2 \times 0.2\,\mathrm{m}</math> and have a thickness of <math>t = 0.9144\,\mathrm{mm}</math>; the receiving cavity is <math>0.2\,\mathrm{m}</math> wide. The outer plate is excited by a uniform pressure load and the resulting acoustic pressure in the receiving cavity is measured at the middle of the rigid wall opposite to the elastic plate (<math>P_1</math> in the sketch).

The following material parameters have been considered for the brass plates and the acoustic fluid:
{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Part
|Parameter
|Value
|Unit
|-style="background-color:#FFFFFF;"
|Brass plates
|<math>E</math>
|<math>104</math>
|<math>\mathrm{GPa}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\rho</math>
|<math>8500</math>
|<math>\mathrm{kg}\,\mathrm{m}^{-3}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\nu</math>
|<math>0.37</math>
|<math>-</math>
|-
|Acoustic fluid
|<math>c</math>
|<math>343</math>
|<math>\mathrm{m}\,\mathrm{s}^{-1}</math>
|-
|
|<math> \rho</math>
|<math> 1.21</math>
|<math> \mathrm{kg}\,\mathrm{m}^{-3}</math>
|-
|}

==Dimensions==
System structure:

:<math>
\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>M \in \mathbb{R}^{n \times n}</math>,
<math>E \in \mathbb{R}^{n \times n}</math>,
<math>K \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=95\,480</math>.

Proportional damping, i.e. <math>E=\alpha M + \beta K</math>, with <math>\alpha=0, \beta=1\cdot 10^{-7}</math> is considered.
The two-way coupling between the structure and the acoustic fluid results in non-symmetric matrices <math>M, E, K</math>.

==Data==
The data is available at [https://doi.org/10.5281/zenodo.7300346 Zenodo].

==Remarks==
* The numerical model resembles the experimental data<ref name="guy81"/> in a frequency range from <math>1\,\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].
* The system has unstable eigenvalues. This is common in interior acoustic problems where no damping is assumed for the acoustic fluid<ref name="cool22"/>.
* 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{dataAum22,
author = {Aumann, Q.},
title = {Matrices for a sound transmission problem},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2022,
doi = {10.5281/zenodo.7300346}
}

* 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==

<references>

<ref name="guy81">R. W. Guy. "[https://www.ingentaconnect.com/content/dav/aaua/1981/00000049/00000004/art00010 The Transmission of Airborne Sound through a Finite Panel, Air Gap, Panel and Cavity Configuration – a Steady State Analysis ]", Acta Acustica united with Acustica, 49(4): 323--333, 1981.</ref>

<ref name="cool22">V. Cool, S. Jonckheere, E. Deckers, W. Desmet. "[http://dx.doi.org/10.1016/j.jsv.2022.116922 Black box stability preserving reduction techniques in the Loewner framework for the efficient time domain simulation of dynamical systems with damping treatments]", Journal of Sound and Vibration, 529: 116922, 2022.</ref>

<ref name="aumann23">Q. Aumann, S. W. R. Werner. "[https://doi.org/10.1016/j.jsv.2022.117363 Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods]", Journal of Sound and Vibration, 543: 117363, 2023.</ref>

</references>

International Space Station

2023-02-17T17:34:57Z

Aumann: Fix bibtex entry

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:Sparse]]
[[Category:MIMO]]

==Description: Components of the International Space Station==

These benchmarks model components of the [[wikipedia:International_Space_Station|International Space Station (ISS)]].
More details can be found in <ref name="gugercin01"/>, <ref name="antoulas01"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

===Component 1R===
This model describes the [[wikipedia:Zvezda_(ISS_module)|Russian Service Module]] (ISS-1R).

===Component 12A===
This model describes the [[wikipedia:Integrated_Truss_Structure#P3/P4,_S3/S4_truss_assemblies|Solar Arrays P3/P4]] (ISS-12A).

==Origin==

This benchmark is part of the '''SLICOT Benchmark Examples for Model Reduction'''<ref name="chahlaoui05"/>.

==Data==

The system matrices <math>A</math>, <math>B</math>, <math>C</math> for the ISS-1R model are available from the [http://slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [http://slicot.org/objects/software/shared/bench-data/iss.zip iss.zip],
the system matrices for the ISS-12A model are available here: [[Media:iss12a.zip|iss12a.zip]].
Both are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] files.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
\dot{x}(t) &=& Ax(t) + Bu(t) \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>A \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times 3}</math>,
<math>C \in \mathbb{R}^{3 \times 270}</math>.

with <math>N=270</math> for ISS-1R and <math>N=1412</math> for ISS-12A.

==Citation==

To cite this benchmark, use the following references:

* For the ISS-1R benchmark itself and its data:
::Niconet e.V., '''SLICOT - Subroutine Library in Systems and Control Theory''', http://www.slicot.org

@MANUAL{slicot_iss,
title = {{SLICOT} - Subroutine Library in Systems and Control Theory},
organization = {Niconet e.V.},
address = <nowiki>{\url{http://www.slicot.org}}</nowiki>,
key = {SLICOT}
}

* For the ISS-12A benchmark and background on the benchmarks:

@INPROCEEDINGS{morGugAB01,
author = <nowiki>{S. Gugercin and A. Antoulas and M. Bedrossian}</nowiki>,
title = {Approximation of the International Space Station 1R and 12A flex models},
booktitle = {Proceedings of the IEEE Conference on Decision and Control},
pages = {1515--1516},
year = {2001},
doi = {10.1109/CDC.2001.981109}
}

==References==

<references>

<ref name="antoulas01"> A.C. Antoulas, D.C. Sorensen and S. Gugercin. [https://doi.org/10.1090/conm/280 A survey of model reduction methods for large-scale systems]. Contemporary Mathematics, 280: 193--219, 2001.</ref>

<ref name="gugercin01">S. Gugercin, A. Antoulas and M. Bedrossian [https://doi.org/10.1109/CDC.2001.981109 Approximation of the International Space Station 1R and 12A flex models]. In: Proceedings of the IEEE Conference on Decision and Control: 1515--1516, 2001.</ref>

<ref name="chahlaoui02"> Y. Chahlaoui, P. Van Dooren, [http://eprints.maths.manchester.ac.uk/1040/1/ChahlaouiV02a.pdf A collection of Benchmark examples for model reduction of linear time invariant dynamical systems], Working Note 2002-2: 2002.</ref>

<ref name="chahlaoui05"> Y. Chahlaoui, P. Van Dooren, [https://doi.org/10.1007/3-540-27909-1_24 Benchmark Examples for Model Reduction of Linear Time-Invariant Dynamical Systems], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 379--392, 2005.</ref>

</references>

Plate with tuned vibration absorbers

2023-01-05T17:24:08Z

Aumann:

{{preliminary}} 
[[Category:benchmark]]
[[Category:linear]]
[[Category:second differential order]]
[[Category:SISO]]
[[Category:MIMO]]

<figure id="fig:plot1">
[[File:Plate_tva.png|480px|thumb|right|<caption>Sketch of the geometry.</caption>]]
</figure>

<figure id="fig:tf">
[[File:Plate_tva_frf.png|480px|thumb|right|<caption>Frequency response function.</caption>]]
</figure>

==Description==
The '''Plate with tuned vibration absorbers''' benchmark models the vibration response of a plate excited by a single harmonic load. The plate is equipped with 108 tuned vibration absorbers (TVA), which change the vibration pattern of the host structure in a narrow frequency band around their tuning frequency<ref name="sun95"/>. Such systems have, for example, been examined in <ref name="jagodzinski20"/> and <ref name="claeys16"/>.

This benchmark models an aluminum plate with dimensions <math>0.8 \times 0.8\,\mathrm{m}</math> and a thickness of <math>t = 1\,\mathrm{mm}</math>. The surrounding edges are simply supported. The plate's surface is equipped with six struts along the <math>y</math>-direction, on which the TVAs are placed. The TVAs have a combined mass of <math>10\,\%</math> of the plate mass and are tuned to <math>f=48\,\mathrm{Hz}</math>. The plate is excited by a single load near one of the corners of the plate (see sketch). The root mean square of the displacement in <math>z</math>-direction at all points of the plate surface is plotted in Figure 2. The effect of the TVAs is clearly visible in the frequency range around their tuning frequency.

The following material parameters have been considered for aluminum:
{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Parameter
|Value
|Unit
|-style="background-color:#FFFFFF;"
|<math>E</math>
|<math>69</math>
|<math>\mathrm{GPa}</math>
|-style="background-color:#FFFFFF;"
|<math>\rho</math>
|<math>2650</math>
|<math>\mathrm{kg}\,\mathrm{m}^{-3}</math>
|-style="background-color:#FFFFFF;"
|<math>\nu</math>
|<math>0.22</math>
|<math>-</math>
|-
|}

==Dimensions==
System structure:

:<math>
\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>M \in \mathbb{R}^{n \times n}</math>,
<math>E \in \mathbb{R}^{n \times n}</math>,
<math>K \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times 1}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
with <math>n=201\,900</math> and <math>q=28\,278</math>.

Proportional damping, i.e. <math>E=\alpha M + \beta K</math>, with <math>\alpha=1\cdot 10^{-2}, \beta=1\cdot 10^{-4}</math> is considered.
The matrices <math>M, E, K</math> are positive (semi-) definite.

==Data==
The data is available at [https://doi.org/10.5281/zenodo.7507012 Zenodo].

==Remarks==
* The dataset also contains a version with a single output (SISO). Here, the displacement of the plate at the location of the load is evaluated.
* The frequency response in the range <math>1\,\mathrm{Hz}</math> to <math>250\,\mathrm{Hz}</math> is 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 a plate with tuned vibration absorbers},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2023,
doi = {10.5281/zenodo.7507012}
}

* 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==

<references>

<ref name="sun95"> J. Q. Sun, M. R. Jolly, M. A. Norris. "[http://dx.doi.org/10.1115/1.2836462 Passive, Adaptive and Active Tuned Vibration Absorbers—A Survey]", Journal of Mechanical Design, 117.B: 234–242, 1995.</ref>

<ref name="jagodzinski20"> D. J. Jagodzinski, M. Miksch, Q. Aumann, G. Müller. "[http://dx.doi.org/10.1080/15397734.2020.1787842 Modeling and optimizing an acoustic metamaterial to minimize low-frequency structure-borne sound]", Mechanics Based Design of Structures and Machines, 50(8): 2877–2891, 2020.</ref>

<ref name="claeys16"> C. Claeys, E. Deckers, B. Pluymers, W. Desmet. "[http://dx.doi.org/10.1016/j.ymssp.2015.08.029 A lightweight vibro-acoustic metamaterial demonstrator: Numerical and experimental investigation]", Mechanical Systems and Signal Processing, 70-71: 853–880, 2016.</ref>

<ref name="aumann23">Q. Aumann, S. W. R. Werner. "[https://doi.org/10.1016/j.jsv.2022.117363 Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods]", Journal of Sound and Vibration, 543: 117363, 2023.</ref>

</references>

File:Plate tva frf.png

2023-01-05T17:23:21Z

Aumann:

Plate with tuned vibration absorbers

2023-01-05T17:23:04Z

Aumann:

{{preliminary}} 
[[Category:benchmark]]
[[Category:linear]]
[[Category:second differential order]]
[[Category:SISO]]
[[Category:MIMO]]

<figure id="fig:plot1">
[[File:Plate_tva.png|480px|thumb|right|<caption>Sketch of the geometry.</caption>]]
</figure>

<figure id="fig:tf">
[[File:plate_tva_frf.png|480px|thumb|right|<caption>Frequency response function.</caption>]]
</figure>

==Description==
The '''Plate with tuned vibration absorbers''' benchmark models the vibration response of a plate excited by a single harmonic load. The plate is equipped with 108 tuned vibration absorbers (TVA), which change the vibration pattern of the host structure in a narrow frequency band around their tuning frequency<ref name="sun95"/>. Such systems have, for example, been examined in <ref name="jagodzinski20"/> and <ref name="claeys16"/>.

This benchmark models an aluminum plate with dimensions <math>0.8 \times 0.8\,\mathrm{m}</math> and a thickness of <math>t = 1\,\mathrm{mm}</math>. The surrounding edges are simply supported. The plate's surface is equipped with six struts along the <math>y</math>-direction, on which the TVAs are placed. The TVAs have a combined mass of <math>10\,\%</math> of the plate mass and are tuned to <math>f=48\,\mathrm{Hz}</math>. The plate is excited by a single load near one of the corners of the plate (see sketch). The root mean square of the displacement in <math>z</math>-direction at all points of the plate surface is plotted in Figure 2. The effect of the TVAs is clearly visible in the frequency range around their tuning frequency.

The following material parameters have been considered for aluminum:
{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Parameter
|Value
|Unit
|-style="background-color:#FFFFFF;"
|<math>E</math>
|<math>69</math>
|<math>\mathrm{GPa}</math>
|-style="background-color:#FFFFFF;"
|<math>\rho</math>
|<math>2650</math>
|<math>\mathrm{kg}\,\mathrm{m}^{-3}</math>
|-style="background-color:#FFFFFF;"
|<math>\nu</math>
|<math>0.22</math>
|<math>-</math>
|-
|}

==Dimensions==
System structure:

:<math>
\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>M \in \mathbb{R}^{n \times n}</math>,
<math>E \in \mathbb{R}^{n \times n}</math>,
<math>K \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times 1}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
with <math>n=201\,900</math> and <math>q=28\,278</math>.

Proportional damping, i.e. <math>E=\alpha M + \beta K</math>, with <math>\alpha=1\cdot 10^{-2}, \beta=1\cdot 10^{-4}</math> is considered.
The matrices <math>M, E, K</math> are positive (semi-) definite.

==Data==
The data is available at [https://doi.org/10.5281/zenodo.7507012 Zenodo].

==Remarks==
* The dataset also contains a version with a single output (SISO). Here, the displacement of the plate at the location of the load is evaluated.
* The frequency response in the range <math>1\,\mathrm{Hz}</math> to <math>250\,\mathrm{Hz}</math> is 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 a plate with tuned vibration absorbers},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2023,
doi = {10.5281/zenodo.7507012}
}

* 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==

<references>

<ref name="sun95"> J. Q. Sun, M. R. Jolly, M. A. Norris. "[http://dx.doi.org/10.1115/1.2836462 Passive, Adaptive and Active Tuned Vibration Absorbers—A Survey]", Journal of Mechanical Design, 117.B: 234–242, 1995.</ref>

<ref name="jagodzinski20"> D. J. Jagodzinski, M. Miksch, Q. Aumann, G. Müller. "[http://dx.doi.org/10.1080/15397734.2020.1787842 Modeling and optimizing an acoustic metamaterial to minimize low-frequency structure-borne sound]", Mechanics Based Design of Structures and Machines, 50(8): 2877–2891, 2020.</ref>

<ref name="claeys16"> C. Claeys, E. Deckers, B. Pluymers, W. Desmet. "[http://dx.doi.org/10.1016/j.ymssp.2015.08.029 A lightweight vibro-acoustic metamaterial demonstrator: Numerical and experimental investigation]", Mechanical Systems and Signal Processing, 70-71: 853–880, 2016.</ref>

<ref name="aumann23">Q. Aumann, S. W. R. Werner. "[https://doi.org/10.1016/j.jsv.2022.117363 Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods]", Journal of Sound and Vibration, 543: 117363, 2023.</ref>

</references>

File:Plate tva.png

2023-01-05T17:14:57Z

Aumann:

Plate with tuned vibration absorbers

2023-01-05T17:00:49Z

Aumann: Created page with "{{preliminary}}  Category:benchmark Category:linear Category:second differential order Category:SISO Category:MIMO <figure id="fig:p..."

{{preliminary}} 
[[Category:benchmark]]
[[Category:linear]]
[[Category:second differential order]]
[[Category:SISO]]
[[Category:MIMO]]

<figure id="fig:plot1">
[[File:plate_tva.png|480px|thumb|right|<caption>Sketch of the geometry.</caption>]]
</figure>

<figure id="fig:tf">
[[File:plate_tva_frf.png|480px|thumb|right|<caption>Frequency response function.</caption>]]
</figure>

==Description==
The '''Plate with tuned vibration absorbers''' benchmark models the vibration response of a plate excited by a single harmonic load. The plate is equipped with 108 tuned vibration absorbers (TVA), which change the vibration pattern of the host structure in a narrow frequency band around their tuning frequency<ref name="sun95"/>. Such systems have, for example, been examined in <ref name="jagodzinski20"/> and <ref name="claeys16"/>.

This benchmark models an aluminum plate with dimensions <math>0.8 \times 0.8\,\mathrm{m}</math> and a thickness of <math>t = 1\,\mathrm{mm}</math>. The surrounding edges are simply supported. The plate's surface is equipped with six struts along the <math>y</math>-direction, on which the TVAs are placed. The TVAs have a combined mass of <math>10\,\%</math> of the plate mass and are tuned to <math>f=48\,\mathrm{Hz}</math>. The plate is excited by a single load near one of the corners of the plate (see sketch). The root mean square of the displacement in <math>z</math>-direction at all points of the plate surface is plotted in Figure 2. The effect of the TVAs is clearly visible in the frequency range around their tuning frequency.

The following material parameters have been considered for aluminum:
{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Parameter
|Value
|Unit
|-style="background-color:#FFFFFF;"
|<math>E</math>
|<math>69</math>
|<math>\mathrm{GPa}</math>
|-style="background-color:#FFFFFF;"
|<math>\rho</math>
|<math>2650</math>
|<math>\mathrm{kg}\,\mathrm{m}^{-3}</math>
|-style="background-color:#FFFFFF;"
|<math>\nu</math>
|<math>0.22</math>
|<math>-</math>
|-
|}

==Dimensions==
System structure:

:<math>
\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>M \in \mathbb{R}^{n \times n}</math>,
<math>E \in \mathbb{R}^{n \times n}</math>,
<math>K \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times 1}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
with <math>n=201\,900</math> and <math>q=28\,278</math>.

Proportional damping, i.e. <math>E=\alpha M + \beta K</math>, with <math>\alpha=1\cdot 10^{-2}, \beta=1\cdot 10^{-4}</math> is considered.
The matrices <math>M, E, K</math> are positive (semi-) definite.

==Data==
The data is available at [https://doi.org/10.5281/zenodo.7507012 Zenodo].

==Remarks==
* The dataset also contains a version with a single output (SISO). Here, the displacement of the plate at the location of the load is evaluated.
* The frequency response in the range <math>1\,\mathrm{Hz}</math> to <math>250\,\mathrm{Hz}</math> is 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 a plate with tuned vibration absorbers},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2023,
doi = {10.5281/zenodo.7507012}
}

* 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==

<references>

<ref name="sun95"> J. Q. Sun, M. R. Jolly, M. A. Norris. "[http://dx.doi.org/10.1115/1.2836462 Passive, Adaptive and Active Tuned Vibration Absorbers—A Survey]", Journal of Mechanical Design, 117.B: 234–242, 1995.</ref>

<ref name="jagodzinski20"> D. J. Jagodzinski, M. Miksch, Q. Aumann, G. Müller. "[http://dx.doi.org/10.1080/15397734.2020.1787842 Modeling and optimizing an acoustic metamaterial to minimize low-frequency structure-borne sound]", Mechanics Based Design of Structures and Machines, 50(8): 2877–2891, 2020.</ref>

<ref name="claeys16"> C. Claeys, E. Deckers, B. Pluymers, W. Desmet. "[http://dx.doi.org/10.1016/j.ymssp.2015.08.029 A lightweight vibro-acoustic metamaterial demonstrator: Numerical and experimental investigation]", Mechanical Systems and Signal Processing, 70-71: 853–880, 2016.</ref>

<ref name="aumann23">Q. Aumann, S. W. R. Werner. "[https://doi.org/10.1016/j.jsv.2022.117363 Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods]", Journal of Sound and Vibration, 543: 117363, 2023.</ref>

</references>

Sound transmission through a plate

2023-01-05T16:07:31Z

Aumann:

{{preliminary}} 
[[Category:benchmark]]
[[Category:linear]]
[[Category:second differential order]]
[[Category:SISO]]

<figure id="fig:plot1">
[[File:guy.png|480px|thumb|right|<caption>Sketch of the geometry.</caption>]]
</figure>

<figure id="fig:tf">
[[File:guy_tf.png|480px|thumb|right|<caption>Transfer function.</caption>]]
</figure>

==Description==
The '''Sound transmission through a plate''' benchmark models the radiation of a vibrating plate and the excitation of a structure by an oscillating acoustic fluid. It is based on an experiment by Guy<ref name="guy81"/>.

The system consists of a cuboid acoustic cavity, where one wall is considered a system of two parallel elastic brass plates with a <math>2\,\mathrm{cm}</math> air gap between them; all other walls are considered rigid. The plates measure <math>0.2 \times 0.2\,\mathrm{m}</math> and have a thickness of <math>t = 0.9144\,\mathrm{mm}</math>; the receiving cavity is <math>0.2\,\mathrm{m}</math> wide. The outer plate is excited by a uniform pressure load and the resulting acoustic pressure in the receiving cavity is measured at the middle of the rigid wall opposite to the elastic plate (<math>P_1</math> in the sketch).

The following material parameters have been considered for the brass plates and the acoustic fluid:
{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Part
|Parameter
|Value
|Unit
|-style="background-color:#FFFFFF;"
|Brass plates
|<math>E</math>
|<math>104</math>
|<math>\mathrm{GPa}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\rho</math>
|<math>8500</math>
|<math>\mathrm{kg}\,\mathrm{m}^{-3}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\nu</math>
|<math>0.37</math>
|<math>-</math>
|-
|Acoustic fluid
|<math>c</math>
|<math>343</math>
|<math>\mathrm{m}\,\mathrm{s}^{-1}</math>
|-
|
|<math> \rho</math>
|<math> 1.21</math>
|<math> \mathrm{kg}\,\mathrm{m}^{-3}</math>
|-
|}

==Dimensions==
System structure:

:<math>
\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>M \in \mathbb{R}^{n \times n}</math>,
<math>E \in \mathbb{R}^{n \times n}</math>,
<math>K \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=95\,480</math>.

Proportional damping, i.e. <math>E=\alpha M + \beta K</math>, with <math>\alpha=0, \beta=1\cdot 10^{-7}</math> is considered.
The two-way coupling between the structure and the acoustic fluid results in non-symmetric matrices <math>M, E, K</math>.

==Data==
The data is available at [https://doi.org/10.5281/zenodo.7300347 Zenodo].

==Remarks==
* The numerical model resembles the experimental data<ref name="guy81"/> in a frequency range from <math>1\,\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].
* The system has unstable eigenvalues. This is common in interior acoustic problems where no damping is assumed for the acoustic fluid<ref name="cool22"/>.
* 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{dataAum22,
author = {Aumann, Q.},
title = {Matrices for a sound transmission problem},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2022,
doi = {10.5281/zenodo.7300347}
}

* 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==

<references>

<ref name="guy81">R. W. Guy. "[https://www.ingentaconnect.com/content/dav/aaua/1981/00000049/00000004/art00010 The Transmission of Airborne Sound through a Finite Panel, Air Gap, Panel and Cavity Configuration – a Steady State Analysis ]", Acta Acustica united with Acustica, 49(4): 323--333, 1981.</ref>

<ref name="cool22">V. Cool, S. Jonckheere, E. Deckers, W. Desmet. "[http://dx.doi.org/10.1016/j.jsv.2022.116922 Black box stability preserving reduction techniques in the Loewner framework for the efficient time domain simulation of dynamical systems with damping treatments]", Journal of Sound and Vibration, 529: 116922, 2022.</ref>

<ref name="aumann23">Q. Aumann, S. W. R. Werner. "[https://doi.org/10.1016/j.jsv.2022.117363 Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods]", Journal of Sound and Vibration, 543: 117363, 2023.</ref>

</references>

Circular Piston

2022-11-18T16:29:37Z

Aumann:

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:second differential order]]

==Description: Axi-Symmetric Infinite Element Model for Circular Piston==
This example is a model of the form

:<math>
M\ddot{x}(t) + E\dot{x}(t) + Kx(t) = B,
</math>

with <math>M</math>, <math>E</math>, and <math>K</math> non-symmetric matrices and <math>M</math> singular.
This is thus a differential algebraic equation.
It is shown that it has index 1<ref name="coyette2003"/>.
The input of the system is <math>B</math>, the output is the state vector <math>x</math>.
The motivation for using model reduction for this type of problems is the reduction of the computation time of a simulation.

This is an example from an acoustic radiation problem discussed in <ref name="pinsky1991"/>.
Consider a circular piston subtending a polar angle <math>0<\theta<\theta_p</math> on a submerged massless and rigid sphere of radius <math>\delta</math>.
The piston vibrates harmonically with a uniform radial acceleration.
The surrounding acoustic domain is unbounded and is characterized by its density <math>\rho</math> and sound speed <math>c</math>.

We denote by <math>p</math> and <math>a_r</math> the prescribed pressure and normal acceleration respectively.
In order to have a steady state solution <math>\tilde{p}(r,\theta,t)</math> verifying

:<math>
\tilde{p}(r,\theta,t) = \mathcal{R} e (p(r,\theta) e^{i\omega t})
</math>

the transient boundary condition is chosen as:

:<math>
a_r = \frac{-1}{\rho} \frac{\partial p(r,\theta)}{\partial r} \big|_{r=a} = \begin{cases} a_0 \sin(\omega t) & 0<\theta<\theta_p \\ 0 & \theta>\theta_p \end{cases}
</math>

The axi-symmetric discrete finite-infinite element model relies on a mesh of linear quadrangle finite elements for the inner domain
(region between spherical surfaces <math>r=\delta</math> and <math>r=1.5\delta</math>).
The numbers of divisions along radial and circumferential directions are <math>5</math> and <math>80</math>, respectively.
The outer domain relies on conjugated infinite elements of order <math>5</math>.
For this example we used <math>\delta=1 [m]</math>, <math>\rho=1.225 [kg/m^3]</math>, <math>c=340 [m/s]</math>, <math>a_0 = 0.001 [m/s^2]</math> and <math>\omega = 1000 [rad/s]</math>.

The matrices <math>K</math>, <math>E</math>, <math>M</math> and the right-hand side <math>f</math> are computed by [http://www.fft.be/ Free Field Technologies].
The dimension of the second-order system is <math>N=2025</math>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38890, see <ref name="bai2005"/>.

==Data==

Download matrices in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/CircularPiston-dim1e3-piston.tar.gz CircularPiston-dim1e3-piston.tar.gz] (1.9 MB).

==Dimensions==

System structure:

:<math>
\begin{align}
M\ddot{x}(t) + E\dot{x}(t) + Kx(t) &= B \\
y(t) &= x(t)
\end{align}
</math>

System dimensions:

<math>M \in \mathbb{R}^{2025 \times 2025}</math>,
<math>E \in \mathbb{R}^{2025 \times 2025}</math>,
<math>K \in \mathbb{R}^{2025 \times 2025}</math>,
<math>B \in \mathbb{R}^{2025 \times 1}</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Circular Piston'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Circular_Piston

@MISC{morwiki_piston,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Circular Piston},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Circular_Piston}</nowiki>,
year = {20XX}
}

* For the background on the benchmark:

@ARTICLE{PinA91,
author = <nowiki>{P.M. Pinsky and N.N. Abboud}</nowiki>,
title = {Finite element solution of the transient exterior structural acoustics problem based on the use of radially asymptotic boundary conditions},
journal = {Computer Methods in Applied Mechanics and Engineering},
volume = {85},
pages = {311--348},
year = {1991},
doi = {10.1016/0045-7825(91)90101-B}
}

==References==

<references>

<ref name="korvink2005"> J.G. Korvink, E.B. Rudnyi, [https://doi.org/10.1007/3-540-27909-1_11 Oberwolfach Benchmark Collection], In: Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 311--315, 2005.</ref>

<ref name="bai2005">Z. Bai, K. Meerbergen, Y. Su, [https://doi.org/10.1007/3-540-27909-1_21 Second Order Models: Linear-Drive Multi-Mode Resonator and Axi Symmetric Model of a Circular Piston]. In: Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol 45: 363--365, 2005.</ref>

<ref name="coyette2003">J.-P. Coyette, K. Meerbergen, M. Robbé, [https://doi.org/10.1002/nme.1419 Time integration for spherical acoustic finite-infinite element models], Numerical Methods in Engineering 64(13): 1752--1768, 2003.</ref>

<ref name="pinsky1991">P.M. Pinsky and N.N. Abboud, [https://doi.org/10.1016/0045-7825(91)90101-B Finite element solution of the transient exterior structural acoustics problem based on the use of radially asymptotic boundary conditions], Computer Methods in Applied Mechanics and Engineering, 85: 311--348, 1991.</ref>

</references>

International Space Station

2022-11-09T14:02:22Z

Aumann:

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:Sparse]]
[[Category:MIMO]]

==Description: Components of the International Space Station==

These benchmarks model components of the [[wikipedia:International_Space_Station|International Space Station (ISS)]].
More details can be found in <ref name="gugercin01"/>, <ref name="antoulas01"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

===Component 1R===
This model describes the [[wikipedia:Zvezda_(ISS_module)|Russian Service Module]] (ISS-1R).

===Component 12A===
This model describes the [[wikipedia:Integrated_Truss_Structure#P3/P4,_S3/S4_truss_assemblies|Solar Arrays P3/P4]] (ISS-12A).

==Origin==

This benchmark is part of the '''SLICOT Benchmark Examples for Model Reduction'''<ref name="chahlaoui05"/>.

==Data==

The system matrices <math>A</math>, <math>B</math>, <math>C</math> for the ISS-1R model are available from the [http://slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [http://slicot.org/objects/software/shared/bench-data/iss.zip iss.zip],
the system matrices for the ISS-12A model are available here: [[Media:iss12a.zip|iss12a.zip]].
Both are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] files.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
\dot{x}(t) &=& Ax(t) + Bu(t) \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>A \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times 3}</math>,
<math>C \in \mathbb{R}^{3 \times 270}</math>.

with <math>N=270</math> for ISS-1R and <math>N=1412</math> for ISS-12A.

==Citation==

To cite this benchmark, use the following references:

* For the ISS-1R benchmark itself and its data:
::Niconet e.V., '''SLICOT - Subroutine Library in Systems and Control Theory''', http://www.slicot.org

@MANUAL{slicot_iss,
title = {{SLICOT} - Subroutine Library in Systems and Control Theory},
organization = {Niconet e.V.},
address = <nowiki>{\url{http://www.slicot.org}}</nowiki>,
key = {SLICOT}
}

* For the ISS-12A benchmark and background on the benchmarks:

@INPROCEEDINGS{morGugAB01,
author = <nowiki>{S. Gugercin, A. Antoulas and M. Bedrossian}</nowiki>,
title = {Approximation of the International Space Station 1R and 12A flex models},
booktitle = {Proceedings of the IEEE Conference on Decision and Control},
pages = {1515--1516},
year = {2001},
doi = {10.1109/CDC.2001.981109}
}

==References==

<references>

<ref name="antoulas01"> A.C. Antoulas, D.C. Sorensen and S. Gugercin. [https://doi.org/10.1090/conm/280 A survey of model reduction methods for large-scale systems]. Contemporary Mathematics, 280: 193--219, 2001.</ref>

<ref name="gugercin01">S. Gugercin, A. Antoulas and M. Bedrossian [https://doi.org/10.1109/CDC.2001.981109 Approximation of the International Space Station 1R and 12A flex models]. In: Proceedings of the IEEE Conference on Decision and Control: 1515--1516, 2001.</ref>

<ref name="chahlaoui02"> Y. Chahlaoui, P. Van Dooren, [http://eprints.maths.manchester.ac.uk/1040/1/ChahlaouiV02a.pdf A collection of Benchmark examples for model reduction of linear time invariant dynamical systems], Working Note 2002-2: 2002.</ref>

<ref name="chahlaoui05"> Y. Chahlaoui, P. Van Dooren, [https://doi.org/10.1007/3-540-27909-1_24 Benchmark Examples for Model Reduction of Linear Time-Invariant Dynamical Systems], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 379--392, 2005.</ref>

</references>

Sound transmission through a plate

2022-11-08T09:52:55Z

Aumann:

{{preliminary}} 
[[Category:benchmark]]
[[Category:linear]]
[[Category:second differential order]]
[[Category:SISO]]

<figure id="fig:plot1">
[[File:guy.png|480px|thumb|right|<caption>Sketch of the geometry.</caption>]]
</figure>

<figure id="fig:tf">
[[File:guy_tf.png|480px|thumb|right|<caption>Transfer function.</caption>]]
</figure>

==Description==
The '''Sound transmission through a plate''' benchmark models the radiation of a vibrating plate and the excitation of a structure by an oscillating acoustic fluid. It is based on an experiment by Guy<ref name="guy81"/>.

The system consists of a cuboid acoustic cavity, where one wall is considered a system of two parallel elastic brass plates with a <math>2\,\mathrm{cm}</math> air gap between them; all other walls are considered rigid. The plates measure <math>0.2 \times 0.2\,\mathrm{m}</math> and have a thickness of <math>t = 0.9144\,\mathrm{mm}</math>; the receiving cavity is <math>0.2\,\mathrm{m}</math> wide. The outer plate is excited by a uniform pressure load and the resulting acoustic pressure in the receiving cavity is measured at the middle of the rigid wall opposite to the elastic plate (<math>P_1</math> in the sketch).

The following material parameters have been considered for the brass plates and the acoustic fluid:
{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Part
|Parameter
|Value
|Unit
|-style="background-color:#FFFFFF;"
|Brass plates
|<math>E</math>
|<math>104</math>
|<math>\mathrm{GPa}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\rho</math>
|<math>8500</math>
|<math>\mathrm{kg}\,\mathrm{m}^{-3}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\nu</math>
|<math>0.37</math>
|<math>-</math>
|-
|Acoustic fluid
|<math>c</math>
|<math>343</math>
|<math>\mathrm{m}\,\mathrm{s}^{-1}</math>
|-
|
|<math> \rho</math>
|<math> 1.21</math>
|<math> \mathrm{kg}\,\mathrm{m}^{-3}</math>
|-
|}

==Dimensions==
System structure:

:<math>
\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>M \in \mathbb{R}^{n \times n}</math>,
<math>E \in \mathbb{R}^{n \times n}</math>,
<math>K \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=95\,480</math>.

Proportional damping, i.e. <math>E=\alpha M + \beta K</math>, with <math>\alpha=0, \beta=1\cdot 10^{-7}</math> is considered.
The two-way coupling between the structure and the acoustic fluid results in non-symmetric matrices <math>M, E, K</math>.

==Data==
The data is available at [https://doi.org/10.5281/zenodo.7300347 Zenodo].

==Remarks==
* The numerical model resembles the experimental data<ref name="guy81"/> in a frequency range from <math>0\,\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].
* The system has unstable eigenvalues. This is common in interior acoustic problems where no damping is assumed for the acoustic fluid<ref name="cool22"/>.
* 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{dataAum22,
author = {Aumann, Q.},
title = {Matrices for a sound transmission problem},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2022,
doi = {10.5281/zenodo.7300347}
}

* 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==

<references>

<ref name="guy81">R. W. Guy. "[https://www.ingentaconnect.com/content/dav/aaua/1981/00000049/00000004/art00010 The Transmission of Airborne Sound through a Finite Panel, Air Gap, Panel and Cavity Configuration – a Steady State Analysis ]", Acta Acustica united with Acustica, 49(4): 323--333, 1981.</ref>

<ref name="cool22">V. Cool, S. Jonckheere, E. Deckers, W. Desmet. "[http://dx.doi.org/10.1016/j.jsv.2022.116922 Black box stability preserving reduction techniques in the Loewner framework for the efficient time domain simulation of dynamical systems with damping treatments]", Journal of Sound and Vibration, 529: 116922, 2022.</ref>

<ref name="aumann23">Q. Aumann, S. W. R. Werner. "[https://doi.org/10.1016/j.jsv.2022.117363 Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods]", Journal of Sound and Vibration, 543: 117363, 2023.</ref>

</references>

Sound transmission through a plate

2022-11-07T18:04:02Z

Aumann: Created page with "{{preliminary}}  Category:benchmark Category:linear Category:second differential order Category:SISO <figure id="fig:plot1"> File:guy...."

{{preliminary}} 
[[Category:benchmark]]
[[Category:linear]]
[[Category:second differential order]]
[[Category:SISO]]

<figure id="fig:plot1">
[[File:guy.png|480px|thumb|right|<caption>Sketch of the geometry.</caption>]]
</figure>

<figure id="fig:tf">
[[File:guy_tf.png|480px|thumb|right|<caption>Transfer function.</caption>]]
</figure>

==Description==
The '''Sound transmission through a plate''' benchmark models the radiation of a vibrating plate and the excitation of a structure by an oscillating acoustic fluid. It is based on an experiment by Guy<ref name="guy81"/>.

The system consists of a cuboid acoustic cavity, where one wall is considered a system of two parallel elastic brass plates with a <math>2\,\mathrm{cm}</math> air gap between them; all other walls are considered rigid. The plates measure <math>0.2 \times 0.2\,\mathrm{m}</math> and have a thickness of <math>t = 0.9144\,\mathrm{mm}</math>; the receiving cavity is <math>0.2\,\mathrm{m}</math> wide. The outer plate is excited by a uniform pressure load and the resulting acoustic pressure in the receiving cavity is measured at the middle of the rigid wall opposite to the elastic plate (<math>P_1</math> in the sketch).

The following material parameters have been considered for the brass plates and the acoustic fluid:
{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Part
|Parameter
|Value
|Unit
|-style="background-color:#FFFFFF;"
|Brass plates
|<math>E</math>
|<math>104</math>
|<math>\mathrm{GPa}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\rho</math>
|<math>8500</math>
|<math>\mathrm{kg}\,\mathrm{m}^{-3}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\nu</math>
|<math>0.37</math>
|<math>-</math>
|-
|Acoustic fluid
|<math>c</math>
|<math>343</math>
|<math>\mathrm{m}\,\mathrm{s}^{-1}</math>
|-
|
|<math> \rho</math>
|<math> 1.21</math>
|<math> \mathrm{kg}\,\mathrm{m}^{-3}</math>
|-
|}

==Dimensions==
System structure:

:<math>
\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t), \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

<math>M \in \mathbb{R}^{n \times n}</math>,
<math>E \in \mathbb{R}^{n \times n}</math>,
<math>K \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=95\,480</math>.

Proportional damping, i.e. <math>E=\alpha M + \beta K</math>, with <math>\alpha=0, \beta=1\cdot 10^{-7}</math> is considered.
The two-way coupling between the structure and the acoustic fluid results in non-symmetric matrices <math>M, E, K</math>.

==Data==
The data is available at [https://doi.org/10.5281/zenodo.7300347 Zenodo].

==Remarks==
* The numerical model resembles the experimental data<ref name="guy81"/> in a frequency range from <math>0\,\mathrm{Hz}</math> to <math>1000\,\mathrm{Hz}</math>.
* The finite element discretization has been performed with [https://github.com/KratosMultiphysics/Kratos Kratos Multiphysics].
* The system has unstable eigenvalues. This is common in interior acoustic problems where no damping is assumed for the acoustic fluid<ref name="cool22"/>.
* 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{dataAum22,
author = {Aumann, Q.},
title = {Matrices for a sound transmission problem},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2022,
doi = {10.5281/zenodo.7300347}
}

* 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==

<references>

<ref name="guy81">R. W. Guy. "[https://www.ingentaconnect.com/content/dav/aaua/1981/00000049/00000004/art00010 The Transmission of Airborne Sound through a Finite Panel, Air Gap, Panel and Cavity Configuration – a Steady State Analysis ]", Acta Acustica united with Acustica, 49(4): 323--333, 1981.</ref>

<ref name="cool22">V. Cool, S. Jonckheere, E. Deckers, W. Desmet. "[http://dx.doi.org/10.1016/j.jsv.2022.116922 Black box stability preserving reduction techniques in the Loewner framework for the efficient time domain simulation of dynamical systems with damping treatments]", Journal of Sound and Vibration, 529: 116922, 2022.</ref>

<ref name="aumann23">Q. Aumann, S. W. R. Werner. "[https://doi.org/10.1016/j.jsv.2022.117363 Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods]", Journal of Sound and Vibration, 543: 117363, 2023.</ref>

</references>

File:Guy tf.png

2022-11-07T16:30:48Z

Aumann:

File:Guy.png

2022-11-07T14:00:42Z

Aumann: