MOR Wiki - User contributions [en]

Windscreen

2023-11-30T09:46:13Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:Linear]]
[[Category:Affine parameter representation]]
[[Category:Parametric]]
[[Category:SISO]]
[[Category:Sparse]]

{{Infobox
|Title = Windscreen
|Benchmark ID = windscreen_n22692m1q1
|Category = oberwolfach
|System-Class = LTI-SOS
|nstates = 22692
|ninputs = 1
|noutputs = 1
|nparameters = 0
|components = B, C, K, M
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
* [[User:Mlinaric]]
* [[User:Yue]]
|Zenodo-link = NA
}}

==Description==
<figure id="fig1">[[File:Windscreen1.gif|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Windscreen2.png|490px|thumb|right|Figure 2]]</figure>

This is an example for a model in the frequency domain of the form

:<math>
\begin{align}
K x + \omega^2 M x & = B \\
y & = C x
\end{align}
</math>

where <math>B</math> represents a unit point load in one unknown of the state vector, <math>C = B^T </math>,
<math>M</math> is a symmetric positive-definite matrix, and <math>K = (1+i\gamma) \widetilde{K}</math> with <math>\widetilde{K}</math> symmetric positive semidefinite.

The test problem is a structural model of a car windscreen. <ref name="meerbergen2007"/>
This is a 3D problem discretized with <math>7564</math> nodes and <math>5400</math> linear hexahedral elements (3 layers of <math>60 \times 30</math> elements).
The mesh is shown in Fig. 1.
The material is glass with the following properties:
The [[wikipedia:Young's_modulus|Young modulus]] is <math>7\times10^{10}\mathrm{N}/\mathrm{m}^2</math>, the density is <math>2490 \mathrm{kg}/\mathrm{m}^3</math>, and the [[wikipedia:Poisson's_ratio|Poisson ratio]] is <math>0.23</math>. The natural damping is <math>10\%</math>, i.e. <math>\gamma=0.1</math>.
The structural boundaries are free (free-free boundary conditions).
The windscreen is subjected to a point force applied on a corner.
The goal of the model reduction is the fast evaluation of <math>y</math>.
Model reduction is used as a fast linear solver for a sequence of parametrized linear systems.

The discretized problem has dimension <math>n=22692</math>.
The goal is to estimate <math>x(\omega)</math> for <math>\omega\in[0.5,200]</math>.
In order to generate the plots, the frequency range was discretized as <math>\{\omega_1,\ldots,\omega_m\} =
\{0.5j,j=1,\ldots,m\}</math> with <math>m=400</math>.

Fig. 1 shows the mesh of the car windscreen and Fig. 2 the frequency response <math>\vert \Re(y(\omega)) \vert</math>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38886.

==Data==

Download matrices in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format:

* [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Windscreen-dim1e4-windscreen.tar.gz Windscreen-dim1e4-windscreen.tar.gz] (21.5 MB)

The archive contains files <tt>windscreen.K</tt>, <tt>windscreen.M</tt> and <tt>windscreen.B</tt> representing <math>K</math>, <math>M</math> and <math>B</math> accordingly.

==Dimensions==

System structure:
:<math>
\begin{align}
(K + \omega^2 M) x & = B \\
y & = C x
\end{align}
</math>
with <math>\omega \in [0.5, 200]</math>.

System dimensions:

<math>K \in \mathbb{C}^{22692 \times 22692}</math>,
<math>M \in \mathbb{R}^{22692 \times 22692}</math>,
<math>B \in \mathbb{R}^{22692 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 22692}</math>,

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

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection, '''Windscreen'''. hosted at MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Windscreen

@MISC{morwiki_windscreen,
author = <nowiki>{{Oberwolfach Benchmark Collection}}</nowiki>,
title = {Windscreen},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Windscreen}</nowiki>,
year = 20XX
}

* For the background on the benchmark:

@article{Mee07,
author = {K. Meerbergen},
title = {Fast frequency response computation for {R}ayleigh damping},
journal = {International Journal for Numerical Methods in Engineering},
volume = {73},
number = {1},
pages = {96--106},
year = {2007},
doi = {10.1002/nme.2058},
}

==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="meerbergen2007"> K. Meerbergen, [https://doi.org/10.1002/nme.2058 Fast frequency response computation for Rayleigh damping], International Journal for Numerical Methods in Engineering, 73(1): 96--106, 2007.</ref>

</references>

Vertical Stand

2023-11-30T09:45:42Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Parametric]]
[[Category:Sparse]]
[[Category:linear]]
[[Category:time varying]]
[[Category:first differential order]]
[[Category:ODE]]
[[Category:MIMO]]
[[Category:CRC-TR-96]]

{{Infobox
|Title = Vertical Stand
|Benchmark ID =
* verticalStandParametric_n16626m6q27
* verticalStandSwitched_n16626m6q27
|Category = misc
|System-Class = AP-LTV-FOS
|nstates = 16626
|ninputs = 6
|noutputs = 27
|nparameters =
* 234
* 11
|components = A, B, C, E
|License = NA
|Creator = [[User:Lnor]]
|Editor =
* [[User:Lnor]]
* [[User:Saak]]
* [[User:Himpe]]
|Zenodo-link = NA
}}

__NUMBEREDHEADINGS__

==Description==

<figure id="fig:cad">
[[File:Staender_Geom_white.jpg|180px|thumb|right|<caption>CAD Geometry</caption>]]
</figure>

The '''vertical stand''' (see Fig. 1) represents a structural part of a machine tool. A pair of guide rails is located on one of the surfaces of this structural part,
and during the machining process, a tool slide is moved to different positions along these rails. The machining process produces a certain amount of heat which is transported through the slide structure into the '''vertical stand'''. This heat source is considered to be a temperature input <math>q_{th}(t)</math> at the guide rails. The induced temperature field, denoted by <math> T </math> is modeled by the [[wikipedia:heat equation|heat equation]]

:<math>
c_p\rho\frac{\partial{T}}{\partial{t}}-\Delta T=0
</math>

with the boundary conditions

:<math>
\lambda\frac{\partial T}{\partial n}=q_{th}(t) \qquad\qquad\qquad
</math> on <math> \Gamma_{rail} </math> (surface where the tool slide is moving on the guide rails),

describing the heat transfer between the tool slide and the '''vertical stand'''.
The heat transfer to the ambiance is given by the locally fixed [[wikipedia:Robin_boundary_condition|Robin-type boundary condition]]

:<math>
\lambda\frac{\partial T}{\partial n}=\kappa_i(T-T_i^{ext})
</math> on <math> \Gamma_{amb} </math> (remaining boundaries).

The motion driven temperature input and the associated change in the temperature field <math>T</math> lead to deformations <math>u</math> within the stand structure. Further, it is assumed that no external forces <math>q_{el}</math> are induced to the system, such that the deformation is purely driven by the change of temperature. Since the mechanical behavior of the machine stand is much faster than the propagation of the thermal field, it is sufficient to consider the stationary [[wikipedia:Linear_elasticity|linear elasticity]] equations
:<math>
\begin{align}
-\operatorname{div}(\sigma(u)) &=q_{el}=0&\text{ on }\Omega,\\
\varepsilon(u) &= {\mathbf{C}}^{-1}:\sigma(u)+\beta(T-T_{ref})I_d&\text{ on }\Omega,\\
{\mathbf{C}}^{-1}\sigma(u) &=\frac{1+\nu}{E_u}\sigma(u)-\frac{\nu}{E_u}\text{tr}(\sigma(u))I_d&\text{ on }\Omega,\\
\varepsilon(u) &= \frac{1}{2}(\nabla u+\nabla u^T)&\text{on }\Omega.
\end{align}
</math>

'''Geometrical dimensions:'''

Stand: Width (<math>x</math> direction): <math>519mm</math>, Height (<math>y</math> direction): <math>2\,010mm</math>, Depth (<math>z</math> direction): <math>480mm</math>

Guide rails: <math>y\in [519, 2\,004] mm</math>

Slide: Width: <math>430 mm</math>, Height: <math>500mm</math>, Depth: <math>490 mm</math>

===Discretized Model===
The solid model has been generated and meshed in [[wikipedia:ANSYS|ANSYS]].
For the spatial discretization the [[wikipedia:finite element method|finite element method]] with linear Lagrange elements has been used and is implemented in [[wikipedia:FEniCS Project|FEniCS]]. The resulting system of [[wikipedia:ordinary differential equations|ordinary differential equations (ODE)]], representing the thermal behavior of the stand, reads

:<math>
\begin{align}
E \frac{\partial}{\partial t} T(t) &= A(t)T(t) + B(t)z(t), \\
T(0) &= T_0,
\end{align}
</math>

with <math>t>0</math> and a system dimension of <math>n=16\,626</math> degrees of freedom and <math>m=6</math> inputs. Note that <math>A(.)\in\mathbb{R}^{n\times n}</math> and <math>B(.)\in\mathbb{R}^{n\times m}</math> are time-dependent matrix-valued functions. That is, the underlying model is represented by a linear time-Varying (LTV) state-space system. More precisely, here the time dependence originates from the change of the boundary condition on <math>\Gamma_{rail}</math> due to the motion of the tool slide. The system input is, according to the boundary conditions, given by
:<math>
z_i=\begin{cases}
q_{th}(t), i=1,\\
\kappa_i T_i^{ext}(t), i=2,\dots,6
\end{cases}.
</math>

The discretized stationary elasticity model becomes
:<math>
Mu(t)=KT(t).
</math>
For the observation of the displacements in single points/regions of interest an output equation of the form
:<math>
y(t) = C u(t)
</math>
is given.

Exploiting the one-sided coupling of the temperature and deformation fields, and reorganizing the elasticity equation in the form <math>u(t)=M^{-1}KT(t)</math>, the heat equation and the elasticity model can easily be combined via the output equation. Finally, the thermo-elastic control system is of the form
:<math>
\begin{align}
E \frac{\partial}{\partial t} T(t) &= A(t)T(t) + B(t)z(t), \\
y(t)& = \tilde{C}T(t),\\
T(0) &= T_0,
\end{align}
</math>
where the modified output matrix <math>\tilde{C}=CM^{-1}KT(t)</math> includes the entire elasticity information.

The motion of the tool slide and the associated variation of the affected input boundary are modeled by two different system representations. The following specific model representations have been developed and investigated in <ref name="morLanSB14" />, <ref name="LanSB15" />, <ref name="Lan17" />.

===Switched Linear System===
<figure id="fig:segm">
[[File:Slide_stand_scheme_new.pdf|thumb|right|220px|<caption>Schematic segmentation</caption>]]
</figure>

For the model description as a switched linear system, the guide rails of the machine stand are modeled as fifteen equally distributed horizontal segments with a height of <math>99mm</math> (see a schematic depiction in Fig. 2). Any of these segments is assumed to be completely covered by the tool slide if its midpoint (in y-direction) lies within the height of the slide. On the other hand, each segment whose midpoint is not covered is treated as not in contact and therefore the slide always covers five to six segments at each time. Still, the covering of six segments does not have a significant effect on the behavior of the temperature and displacement fields. Due to that and in order to keep the number of subsystems small, this scenario will be neglected. Then, in fact eleven distinct, discrete boundary condition configurations for the stand model that are prescribed by the geometrical dimensions of the segmentation and the tool slide are defined. These distinguishable setups define the subsystems
:<math>
\begin{align}
E\dot{T}(t)&=A_{\alpha}T(t)+B_{\alpha}z(t),\\
y(t)&=\tilde{C}T(t),
\end{align}
</math>
of the switched linear system <ref name=Lib03/>, where <math>\alpha</math> is a piecewise constant function of time, which takes its value from the index set <math>\mathcal{J}=\{1,\dots,11\}</math>. To be more precise, the switching signal <math>\alpha</math> implicitly maps the slide position to the number of the currently active subsystem.

===Linear Parameter-Varying System===
For the parametric model description, the finite element nodes located at the guide rails are clustered with respect to their y-coordinates. This results in <math>233</math> distinct layers in y-direction. According to these layers, the matrices <math>A(t)=A(\mu(t)), B(t)=B(\mu(t)) </math> are defined in a parameter-affine representation of the form
:<math>
\begin{align}
A(\mu) = A_0+f_1(\mu)A_1+...+f_{m_A}(\mu)A_{m_A},\\
B(\mu) = B_0+g_1(\mu)B_1+...+g_{m_B}(\mu)B_{m_B}
\end{align}
</math>
with the scalar functions <math>f_i,g_j\in\{0,1\}, i=1,...,m_A, j=1,...,m_B</math> selecting the active layers, covered by the tool slide and <math>\mu(t)</math> being the position of the middle point (vertical / y-direction) of the slide. The matrix <math>A_0\in\mathbb{R}^{n\times n}</math> consists of the discretization of the [[wikipedia:Laplace operator|Laplacian]] <math>\Delta</math>, as well as the discrete portions from the Robin-type boundaries that correspond to the temperature exchange with the ambiance. The remaining summands <math>A_j\in\mathbb{R}^{n\times n},~j=1,...,m_A</math> denote the discretization associated to the moving Robin-type boundaries. For the representation of the input matrix, the summand <math>B_0\in\mathbb{R}^{n\times 6}</math> consists of a single zero column followed by five columns related to the inputs <math>z_i,~i=2,...,6</math>. The remaing matrices <math>B_j,~j=1,...,m_B</math> are built by a single column corresponding to the different layers followed by a zero block of dimension <math>n\times 5</math> designated to fit the dimension of <math>B_0</math>.

Note that in general, the number of summands of these representations need not be equal. Still, according to the number of layers, for this example, it holds that <math>m_A=m_B=233</math>. For more details on parametric models, see e.g., <ref name="morBauBBetal11" /> and the references therein.

Then, the final linear parameter-varying (LPV) reformulation of the above LTV system reads
:<math>
\begin{align}
E\dot{T}(t)&=A(\mu)T(t)+B(\mu)z(t),\\
y(t)&=\tilde{C}T(t).
\end{align}
</math>

==Acknowledgement & Origin==
The base model was developed <ref name="GalGM11" />, <ref name="GalGM15" /> in the [http://transregio96.de Collaborative Research Centre Transregio 96] ''Thermo-Energetic Design of Machine Tools'' funded by the [http://www.dfg.de/en/index.jsp Deutsche Forschungsgemeinschaft] .

==Data==
====Switched System Data====
The data file [[Media:VertStand_SLS.tar.gz|VertStand_SLS.tar.gz]] contains the matrices


:<math>
\begin{align}
E\in\mathbb{R}^{n\times n}, \tilde{C}\in\mathbb{R}^{q\times n}, A_\alpha\in\mathbb{R}^{n\times n}, B_\alpha\in\mathbb{R}^{n\times m}, \alpha=1,\dots,11.
\end{align}
</math>
defining the subsystems of the switched linear system.
The matrices <math>A_\alpha</math> and <math>B_\alpha</math> are numbered according to the slide position in descending order (1 - uppermost slide position / 11 - lowest slide position).

System dimensions:
:<math>
n=16\,626, m=6, q=27
</math>

====Parametric System Data====
The data file [[Media:VertStand_PAR.tar.gz|VertStand_PAR.tar.gz]] contains the matrices

:<math>
E,A_j\in\mathbb{R}^{n\times n},j=1,...,234, B_{rail}\in\mathbb{R}^{n\times 233}, B_{amb}\in\mathbb{R}^{n\times 5}</math> and <math>\tilde{C}\in\mathbb{R}^{q\times n}, q=27, n=16\,626,
</math>
as well as a file ''ycoord_layers.mtx'' containing the y-coordinates of the layers located on the guide rails.

Here <math>B_{rail} </math> contains all columns corresponding to the different layers on the guide rails and <math>B_{amb}</math> correlates to the boundaries where the ambient temperatures act on.

In order to set up the parameter dependent matrices <math>A(\mu),B(\mu)</math> the active matrices <math>A_i</math> and columns <math>B_{rail}(:,i)</math> associated to the covered layers have to be identified by the current position <math>\mu(t)</math> (vertical middle point of the slide) and the geometrical dimensions of the tool slide and the y-coordinates of the different layers given in ''ycoord_layers.mtx''. Then, <math>B(\mu)</math> has to be set up in the form <math>B(\mu)=[\sum_{i\in id_{active}}\!\!\!\!\!B_{rail}(:,i),B_{amb}]</math>, where <math>id_{active}</math> denotes the set of covered layers and their corresponding columns in <math>B_{rail}</math>.



The matrices have been re-indexed (starting with 1) for [[MORB]].

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_vertstand,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Vertical Stand},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Vertical_Stand}</nowiki>,
year = 2014
}

* For the background on the benchmark:

@Article{morLanSB14,
author = {Lang, N. and Saak, J. and Benner, P.},
title = {Model Order Reduction for Systems with Moving Loads},
journal = {at-Automatisierungstechnik},
volume = 62,
number = 7,
pages = {512--522},
year = 2014,
doi = {10.1515/auto-2014-1095}
}

==References==

<references>

<ref name="morBauBBetal11"> U. Baur, C. A. Beattie, P. Benner, and S. Gugercin,
"[http://epubs.siam.org/doi/abs/10.1137/090776925 Interpolatory projection methods for parameterized model reduction]",
SIAM J. Sci. Comput., 33(5):2489-2518, 2011</ref>

<ref name="Lib03">D. Liberzon, [https://doi.org/10.1007/978-3-319-12625-8_8 Switching in Systems and Control] , Springer-Verlag, New York, 2003</ref>

<ref name="GalGM11"> A. Galant, K. Großmann, and A. Mühl, Model Order Reduction (MOR) for
Thermo-Elastic Models of Frame Structural Components on Machine Tools.
ANSYS Conference & 29th CADFEM Users’ Meeting 2011, October
19-21, 2011, Stuttgart, Germany</ref>.

<ref name="morLanSB14">N. Lang and J. Saak and P. Benner,
[https://doi.org/10.1515/auto-2014-1095 Model Order Reduction for Systems with Moving Loads] ,
in De Gruyter Oldenbourg: at-Automatisierungstechnik, Volume 62, Issue 7, Pages 512-522, 2014</ref>

<ref name="LanSB15">N. Lang, J. Saak and P. Benner, [https://doi.org/10.1007/978-3-319-12625-8_8 Model Order Reduction for Thermo-Elastic Assembly Group Models] , In: Thermo Energetic Design of Machine Tools, Lecture Notes in Production Engineering, 85-92, 2015</ref>

<ref name="Lan17">N. Lang, [https://www.logos-verlag.de/cgi-bin/buch/isbn/4700 Numerical Methods for Large-Scale Linear Time-Varying Control Systems and related Differential Matrix Equations] , Logos-Verlag, 2018. ISBN: 978-3-8325-4700-4</ref>

<ref name="GalGM15">A. Galant, K. Großmann and A. Mühl, [https://doi.org/10.1007/978-3-319-12625-8_7 Thermo-Elastic Simulation of Entire Machine Tool] , In: Thermo Energetic Design of Machine Tools, Lecture Notes in Production Engineering, 69-84, 2015</ref>

</references>

==Contact==

'' [[User:Saak]]''

Tunable Optical Filter

2023-11-30T09:45:20Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:MIMO]]
[[Category:Sparse]]

{{Infobox
|Title = Tunable Optical Filter
|Benchmark ID =
* tunableOpticalFilter_n106437m1q5
* tunableOpticalFilter_n1668m1q5
|Category = oberwolfach
|System-Class = LTI-FOS
|nstates =
* 106437
* 1668
|ninputs = 1
|noutputs = 5
|nparameters = 0
|components = A, B, C, E
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
* [[User:Werner]]
|Zenodo-link = NA
}}

==Description==
<figure id="fig1">[[File:TunOptFilt1.jpg|490px|thumb|right|<caption>Tunable optical filter.</caption>]]</figure>

A benchmark for the heat transfer problem, related to modeling of a tunable optical filter (see Fig. 1), is presented.
It can be used to apply model reduction algorithms to a linear first-order problem.

===Modeling===

The DFG project AFON (funded under grant ZA 276/2-1) aimed at the development of an optical filter, which is tunable by thermal means.
The thin-film filter is configured as a membrane in order to improve thermal isolation.
Fabrication is based on silicon technology. Wavelength tuning is achieved through thermal modulation of resonator optical thickness, using metal resistor deposited onto the membrane.
The devices features low power consumption, high tuning speed and excellent optical performance<ref name="hohlfeld2004"/>.
These tunable devices are an important component in various optical systems such as tunable lasers.

The benchmark contains a simplified thermal model of a filter device.
It helps designers to consider important thermal issues, such as what electrical power should be applied in order to reach the critical temperature at the membrane or homogeneous temperature distribution over the membrane.
The original model is the heat transfer partial differential equation

:<math>
\begin{align}
\nabla \cdot (\kappa(r) \nabla T(r,t)) + Q(r, t) - \rho(r)C_{p}(r)\frac{\partial T(r, t)}{\partial t} & = 0,
\end{align}
</math>

where <math>r</math> is the position, <math>t</math> is the time, <math>\kappa</math> is the thermal conductivity of the material, <math>C_{p}</math> is the specific heat capacity, <math>\rho</math> is the mass density, <math>Q</math> is the heat generation rate that is nonzero only within the heater, and <math>T</math> is the unknown temperature distribution to be determined.

There are two different benchmarks, 2D model and 3D model (see Table [[#tab1|1]]).
Due to modeling differences, their simulation results cannot be compared with each other directly.

{| class="wikitable" style="margin: auto;" id="tab1"
|+ style="caption-side:bottom;"|''Table 1: Tunable optical filter benchmarks.''
|-
|Code
|Comment
|Dimension
|nnz(A)
|nnz(E)
|-
|filter2D
|2D, linear elements, PLANE55
|1668
|6209
|1668
|-
|filter3D
|3D, linear elements, SOLID90
|108373
|1406808
|1406791
|}

===Discretization===

The solid device models have been designed, meshed and discretized in [http://www.ansys.com/ ANSYS] 6.1 by the finite element method.
All material properties are considered as temperature independent.
Temperature is assumed to be in Celsius with the initial state of <math>0^{\circ}C</math>.
The Dirichlet boundary conditions of <math>T = 0^{\circ}C</math> have been applied at the bottom of the chip.
Output nodes for the models are described in Table [[#tab2|2]] and schematically displayed in Fig. 2.
The first output is located at the very center of the membrane.
By simulating its temperature one can prove how much input power is needed to reach the critical membrane temperature for each wavelength.
Furthermore, the outputs 2 to 5 must be very close to output 1 (homogeneous temperature distribution) in order to provide the same optical properties across the complete diameter of the laser beam.

{| class="wikitable" style="margin: auto;" id="tab2"
|+ style="caption-side:bottom;"|''Table 2: Outputs for the optical filter model.''
|-
|Number
|Code
|Comment
|-
|1
|Memb1
|Membrane center
|-
|2
|Memb2
|Membrane node with radius 25E-6, theta 90°
|-
|3
|Memb3
|Membrane node with radius 50E-6 theta 90°
|-
|4
|Memb4
|Membrane node with radius 25E-6, theta 135°
|-
|5
|Memb5
|Membrane node with radius 50E-6 theta 135°
|}

<figure id="fig2">[[File:TunOptFilt2.jpg|490px|thumb|right|<caption>Schematic position of the chosen output nodes.</caption>]]</figure>

The input corresponds to the constant power of of <math>1 mW</math> for 2D model and <math>10 mW</math> for 3D model.
Hence practically, the benchmark contains a constant load vector.
The linear ordinary differential equations of the first order are written as:

:<math>
\begin{align}
E\dot{x}(t) &= Ax(t) + b,\\
y(t) &= Cx(t),
\end{align}
</math>

where <math>E</math> and <math>A</math> are the symmetric sparse system matrices (heat capacity and heat conductivity matrix), <math>b</math> is the load vector, <math>C</math> is the output matrix, and <math>x</math> is the vector of unknown temperatures.

The output of the transient simulation for node 1 over the rise time of the device (<math>0.25 s</math>) for 3D model can be found in <tt>Filter3DTransResults</tt>.
The results can be used to compare the solution of a reduced model with the original one.
The time integration has been performed in ANSYS with accuracy of about <math>0.1\%</math>.
The results are given as matrices where the first row is made of times, the second of the temperatures.

The discussion of electro-thermal modeling related to the benchmark can be found in<ref name="bechthold2005"/>.

==Acknowledgments==
This work is partially funded by the DFG projects '''AFON (ZA 276/2-1), MST-Compact (KO-1883/6)''' and an operating grant of the University of Freiburg.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38875, see: <ref name="hohlfeld2005"/>.

==Data==

This benchmark includes two data sets, one for the 2D and one for 3D model.
The matrices can be downloaded in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format:
* [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/TunableOpticalFilter-dim1e3-filter2D.tar.gz TunableOpticalFilter-dim1e3-filter2D.tar.gz] (104.0 kB)
* [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/TunableOpticalFilter-dim1e5-filter3D.tar.gz TunableOpticalFilter-dim1e5-filter3D.tar.gz] (35.7 MB)
The matrix name is used as an extension of the matrix file.
The file <tt>*.C.names</tt> contains a list of ouput names written consecutively.
The file <tt>Filter3DTransResults</tt> contains the output of the transient simulation for node 1 over the rise time of the device (<math>0.25 s</math>) for the 3D model and <tt>Filter3DTransResults.names</tt> the corresponding output names.
The system matrices have been extracted from ANSYS models by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem].

==Dimensions==

System structure:

:<math>
\begin{align}
E\dot{x}(t) &= Ax(t) + b,\\
y(t) &= Cx(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 1}</math>,
<math>C \in \mathbb{R}^{5 \times N}</math>,

System variants:

<tt>filter2d</tt>: <math>N = 1668</math>,
<tt>filter3d</tt>: <math>N = 108373</math>

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection, '''Tunable Optical Filter'''. hosted at MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Tunable_Optical_Filter

@MISC{morwiki_optical,
author = <nowiki>{{Oberwolfach Benchmark Collection}}</nowiki>,
title = {Tunable Optical Filter},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Tunable_Optical_Filter}</nowiki>,
year = 20XX
}

* For the background on the benchmark:

@ARTICLE{HohZ04,
author = {D. Hohlfeld and H. Zappe},
title = {An all-dielectric tunable optical filter based on the thermo-optic effect},
journal = {Journal of Optics A: Pure and Applied Optics},
volume = {6},
number = {6},
pages = {504--511},
year = {2004},
doi = {10.1088/1464-4258/6/6/002}
}

==References==

<references>

<ref name="hohlfeld2004"> D. Hohlfeld, H. Zappe, [https://doi.org/10.1088/1464-4258/6/6/002 All-dielectric tunable optical filter based on the thermo-optic effect], Journal of Optics A: Pure and Applied Optics, 6(6): 504--511, 2004.</ref>

<ref name="bechthold2005"> T. Bechthold, [https://freidok.uni-freiburg.de/data/1914 Model order reduction of electro-thermal MEMS], Albert-Ludwigs-Universität Freiburg, 2005.</ref>

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

<ref name="hohlfeld2005"> D. Hohlfeld, T. Bechtold, H. Zappe, [https://doi.org/10.1007/3-540-27909-1_15 Tunable Optical Filter], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 337--340, 2005.</ref>

</references>

Transmission Lines

2023-11-30T09:44:52Z

Malhotra: infobox edit

[[Category:benchmark]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:Differential_algebraic_system]]

{{Infobox
|Title = Transmission Lines
|Benchmark ID =
* transmissionLines_n1600m14q14
* transmissionLines_n2624m30q30
* transmissionLines_n5248m62q62
|Category = misc
|System-Class = LTI-FOS
|nstates =
* 1600
* 2624
* 5248
|ninputs =
* 14
* 30
* 62
|noutputs =
* 14
* 30
* 62
|nparameters = 0
|components = A, B, C, E
|License = NA
|Creator = [[User:Deluca]]
|Editor =
* [[User:Deluca]]
* [[User:Himpe]]
|Zenodo-link = NA
}}

==Definition==

In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency,
that is, currents with a frequency high enough that their wave nature must be taken into account.
'''Transmission lines''' are used for purposes such as connecting radio transmitters and receivers with their antennas,
distributing cable television signals, and computer network connections.

In many electric circuits, the length of the wires connecting the components can often be ignored.
That is, the voltage on the wire at a given time can be assumed to be the same at all points.
However, when the voltage changes as fast as the signal travels through the wire,
the length becomes important and the wire must be treated as a transmission line, with distributed parameters.
Stated in another way, the length of the wire is important when the signal includes frequency components with corresponding wavelengths comparable to or less than the length of the wire.

A common rule of thumb is that the cable or wire should be treated as a transmission line if its length is greater than <math>1/10</math> of the wavelength,
and the interconnect is called "electrically long".
At this length the phase delay and the interference of any reflections on the line (as well as other undesired effects) become important and can lead to unpredictable behavior in systems which have not been carefully designed using transmission line theory.



==Description==

The decrease of integrated circuit feature size and the increase of operating frequencies require 3-D electromagnetic methods,
such as the [[wikipedia:Partial_element_equivalent_circuit|Partial Element Equivalent Circuit]] (PEEC) method; it stems from the integral equation form of [[wikipedia:Maxwell's_equations|Maxwell's equations]].
The main difference of the PEEC method with other integral-Equation-based techniques, such as the method of moments, resides in the fact that it provides a circuit interpretation of the [[wikipedia:Electric-field_integral_equation|Electric Field Integral Equation]] (EFIE) in terms of partial elements, namely resistances, partial inductances, and coefficients of potential. In the standard approach, volumes and surfaces are discretized into elementary regions, hexahedra, and patches respectively over which the current and charge densities are expanded into a series of basis functions.
Pulse basis functions are usually adopted as expansion and weight functions.
Such choice of pulse basis functions corresponds to assuming constant current density and charge density over the elementary volume (inductive) and surface (capacitive) cells, respectively.
Following the standard Galerkin's testing procedure, topological elements, namely nodes and branches, are generated and electrical lumped elements are identified modeling both the magnetic and electric field coupling.
Conductors are modeled by their ohmic resistance, while dielectrics requires modeling the excess charge due to the dielectric polarization.
Magnetic and electric field coupling are modeled by partial inductances and coefficients of potential, respectively.

The magnetic field coupling between two inductive volume cells <math>\alpha</math> and <math>\beta</math> is described by the partial inductance

:<math> L_{p_{\alpha\beta}}=\frac{\mu}{4\pi}\frac{1}{a_{\alpha}a_{\beta}}\int_{u_{\alpha}}\int_{u_{\beta}}\frac{1}{R_{\alpha\beta}}\,du_{\alpha}\,du_{\beta} </math>

where <math>R_{\alpha\beta}</math> is the distance between any two points in the volumes <math>u_{\alpha}</math> and <math>u_{\beta}</math> with <math>a_{\alpha}</math> and <math>a_{\beta}</math> their cross section. The electric field coupling between two capacitive surface cells <math>\delta</math> and <math>\gamma</math> is modeled by the coefficient of the potential

:<math> P_{\delta\gamma}=\frac{1}{4\pi\epsilon}\frac{1}{S_{\delta}S_{\gamma}}\int_{S_{\delta}}\int_{S_{\gamma}}\frac{1}{R_{\delta\gamma}}\,dS_{\delta}\,dS_{\gamma} </math>

where <math>R_{\delta\gamma}</math> is the distance between any two points on the surfaces <math>\delta</math> and <math>\gamma</math>, while <math>S_{\delta}</math> and <math>S_{\gamma}</math> denote the area of their respective surfaces <ref name="ferranti11"/>.

Generalized Kirchhoff's laws for conductors, when dielectrics are considered, can be rewritten as

:<math> \textbf{P}^{-1}\frac{d\textbf{v}(t)}{dt}-\textbf{A}^T\textbf{i}(t)+\textbf{i}_e(t)=0, </math>

<figure id="fig:peec">[[File:Peec.jpg|400px|frame|<caption>Illustration of PEEC circuit electrical quantities for a conductor elementary cell (Figure from <ref name="ferranti11"/>).</caption>]]</figure>

:<math> -\textbf{A}\textbf{v}(t)-\textbf{L}_p\frac{d\textbf{i}(t)}{dt}-\textbf{v}_d(t)=0, </math>

:<math> \textbf{i}(t)=\textbf{C}_d\frac{d\textbf{v}_d(t)}{dt} </math>

where <math>\textbf{A}</math> is the connectivity matrix, <math>\textbf{v}(t)</math> denotes the node potentials to infinity, <math>\textbf{i}(t)</math> and <math>\textbf{i}_e(t)</math> represent the currents flowing in volume cells and the external currents, respectively, <math>\textbf{v}_d(t)</math> is the excess capacitance voltage drop, which is related to the excess charge by <math>\textbf{v}_d(t)=\textbf{C}_d^{-1}\textbf{q}_d(t)</math>. A selection matrix <math>\textbf{K}</math> is introduced to define the port voltages by selecting node potentials. The same matrix is used to obtain the external currents <math>\textbf{i}_e(t)</math> by the currents <math>\textbf{i}_s(t)</math>, which are of opposite sign with respect to the <math>n_p</math> port currents <math>\textbf{i}_p(t)</math>,

:<math> \textbf{v}_p(t)=\textbf{K}\textbf{v}(t), </math>

:<math> \textbf{i}_e(t)=\textbf{K}^T\textbf{i}_s(t). </math>

An example of PEEC circuit electrical quantities for a conductor elementary cell is illustrated, in the Laplace domain, in Fig. 1, where the current-controlled voltage sources <math>sL_{p,ij}I_j</math> and the current-controlled current sources <math>I_{cci}</math> model the magnetic and electric coupling, respectively.

Thus, assuming that we are interested in generating an admittance representation having <math>n_p</math> output currents under voltage excitation, and let us denote with <math>n_n</math> the number of nodes, <math>n_i</math> the number of branches where currents flow, <math>n_c</math> the number of branches of conductors, <math>n_d</math> the number of dielectrics, <math>n_d</math> the additional unknowns since dielectrics require the excess capacitance to model the polarization charge, and <math>n_u=n_i+n_d+n_n+n_p</math> the global number of unknowns, and if the [[wikipedia:Modified_nodal_analysis|Modified Nodal Analysis]] (MNA) approach is used, we have:

:::<math> \left[ \begin{array}{cccc} \textbf{P} & \textbf{0}_{n_n,n_i} & \textbf{0}_{n_n,n_d} & \textbf{0}_{n_n,n_p} \\ \textbf{0}_{n_i,n_n} & \textbf{L}_p & \textbf{0}_{n_i,n_d} & \textbf{0}_{n_i,n_p} \\ \textbf{0}_{n_d,n_n} & \textbf{0}_{n_d,n_i} & \textbf{C}_d & \textbf{0}_{n_d,n_p} \\ \textbf{0}_{n_p,n_n} & \textbf{0}_{n_p,n_i} & \textbf{0}_{n_p,n_d} & \textbf{0}_{n_p,n_p} \end{array}\right]\frac{d}{dt}\left[ \begin{array}{c}\textbf{q}(t) \\ \textbf{i}(t) \\ \textbf{v}_d(t) \\ \textbf{i}_s(t) \end{array}\right]= </math>

:<math>= - \left[ \begin{array}{cccc}\textbf{0}_{n_n,n_n} & -\textbf{P}\textbf{A}^T & \textbf{0}_{n_n,n_d} & \textbf{P}\textbf{K}^T \\ \textbf{AP} & \textbf{R} & \Phi & \textbf{0}_{n_i,n_p} \\ \textbf{0}_{n_d,n_n} & -\Phi^T & \textbf{0}_{n_d,n_d} & \textbf{0}_{n_d,n_p} \\ -\textbf{K}\textbf{P} & \textbf{0}_{n_p,n_i} & \textbf{0}_{n_p,n_d} & \textbf{0}_{n_p,n_p} \end{array}\right]\cdot\left[ \begin{array}{c} \textbf{q}(t) \\ \textbf{i}(t) \\ \textbf{v}_d(t) \\ \textbf{i}_s(t) \end{array}\right]+ \left[ \begin{array}{c}\textbf{0}_{n_n+n_i+n_d,n_p} \\ -\textbf{I}_{n_p,n_p} \end{array}\right] \cdot [ \textbf{v}_p(t) ]. </math>

Here <math>\textbf{0}</math> is a matrix of zeros, <math>\textbf{I}</math> is the identity matrix, both are with appropriate dimensions, and <math>\Phi=\left[ \begin{array}{c} \textbf{0}_{n_c,n_d} \\ \textbf{I}_{n_d,n_d} \end{array}\right]</math>. Then, in a more compact form, the above equation can be written as:

:<math> \left\{ \begin{array}{c} \textbf{C}\frac{d\textbf{x}(t)}{dt}=-\textbf{G}\textbf{x}(t)+\textbf{B}\textbf{u}(t)\\
\textbf{i}_p(t)=\textbf{L}^T\textbf{x}(t) \end{array}\right . \qquad (1)</math>

with <math>\textbf{x}(t)=\left[ \begin{array}{cccc} \textbf{q}(t)\quad\textbf{i}(t)\quad\textbf{v}_d(t)\quad\textbf{i}_s(t) \end{array}\right]^T</math>. Since this is an <math>n_p</math>-port formulation, whereby the only sources are the voltage sources at the <math>n_p</math>-ports nodes, <math>\textbf{B}=\textbf{L}</math> where <math>\textbf{B}\in\mathbb R^{n_u\times n_p}</math> (for more details on this model, refer to <ref name="ferranti11"/>).

==Motivation of MOR==

Since the number of equations produced by 3-D electromagnetic method PEEC is usually very large,
the inclusion of the PEEC model directly into a circuit simulator (like [[wikipedia:SPICE|SPICE]]) is computationally intractable for complex structures,
where the number of circuit elements can be tens of thousands.

==Data==

All data sets (in a MATLAB formatted data, downloadable in [[Media:TransmissionLines.rar|TransmissionLines.rar]]) in Fig. 1 are referred to as the multiconductor '''transmission lines''' in a MNA form, coming from the PEEC method (then, with dense matrices since they are obtained from the integral formulation of Maxwell's equation).
The LTI descriptor systems have the form of, equation <math>(1)</math>, where <math>C\in\mathbb R^{n\times n}</math> (with <math>C=C^T\ge0</math>), <math>G\in\mathbb R^{n\times n}</math> (with <math>G+G^T\ge0</math>), <math>B\in\mathbb R^{n\times m}</math> and <math>L=B^T</math>, <math>x(t)\in\mathbb R^n</math> is the vector of variables (charges, currents and node potential), the input signal <math>u(t)\in\mathbb R^m</math> are the sources (current or voltage generators depending on what one wants to analyze:
the impedances or the admittances) linked to some node, the output <math>y(t)\in\mathbb R^m</math> are the observation across the node where the sources are inserted. An accurate model of the dynamics of these data sets is generated between 10 kHz and 20 GHz.

<figtable id="tab:peec">
{| border="1"
! Name of the data set !! Matrices !! Dimension !! Number of inputs
|-
| <tt>dsysPEEC-MTLn1600m14</tt> || <math>G</math>, <math>B</math>, <math>C</math> (<math>L=B^T</math>) || 1600 || 14
|-
| <tt>dsysPEEC-MTLn2624m30</tt> || dss object (*) || 2624 || 30
|-
| <tt>dsysPEEC-MTLn5248m62</tt> || dss object (*) || 5248 || 62
|}
</figtable>

one can extract the matrices with Matlab command:

[G,B,L,D,C] = dssdata(dssObjectName);

e.g., if one wants to work on one of the last two data sets of this table,
just load it into the Matlab Workspace and type the command aforementioned on the Command Windows;
for the first example, once one loads the data, the Workspace shows directly the matrices.
Note that <math>D = 0</math>.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
E\dot{x}(t) &=& Ax(t) + Bu(t) \\
y(t) &=& C x(t)
\end{array}
</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}^{M \times N}</math>.

System variants:

<tt>dsysPEEC-MTLn1600m14</tt>: <math>N = 1600</math>, <math>M = 14</math>,
<tt>dsysPEEC-MTLn2624m30</tt>: <math>N = 2624</math>, <math>M = 30</math>,
<tt>dsysPEEC-MTLn5248m62</tt>: <math>N = 5248</math>, <math>M = 62</math>

==References==

<references>

<ref name="ferranti11"> F. Ferranti, G. Antonini, T. Dhaene, L. Knockaert, and A. E. Ruehli, "[https://doi.org/10.1109/TCPMT.2010.2101912 Physics-based passivity-preserving parameterized model order reduction for PEEC circuit analysis]", IEEE Transactions on Components, Packaging and Manufacturing Technology, 1(3): 399--409, 2011.</ref>

</references>

==Contact==

[[User:Deluca]]

[[User:Feng]]

Transmission Lines

2023-11-30T09:43:32Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:Differential_algebraic_system]]

{{Infobox
|Title = Transmission Lines (SLICOT)
|Benchmark ID = transmissionLines_n256m2q2
|Category = slicot
|System-Class = LTI-FOS
|nstates = 256
|ninputs = 2
|noutputs = 2
|nparameters = 0
|components = A, B, C, E
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
|Zenodo-link = NA
}}

==Definition==

In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency,
that is, currents with a frequency high enough that their wave nature must be taken into account.
'''Transmission lines''' are used for purposes such as connecting radio transmitters and receivers with their antennas,
distributing cable television signals, and computer network connections.

In many electric circuits, the length of the wires connecting the components can often be ignored.
That is, the voltage on the wire at a given time can be assumed to be the same at all points.
However, when the voltage changes as fast as the signal travels through the wire,
the length becomes important and the wire must be treated as a transmission line, with distributed parameters.
Stated in another way, the length of the wire is important when the signal includes frequency components with corresponding wavelengths comparable to or less than the length of the wire.

A common rule of thumb is that the cable or wire should be treated as a transmission line if its length is greater than <math>1/10</math> of the wavelength,
and the interconnect is called "electrically long".
At this length the phase delay and the interference of any reflections on the line (as well as other undesired effects) become important and can lead to unpredictable behavior in systems which have not been carefully designed using transmission line theory.



==Description==

The decrease of integrated circuit feature size and the increase of operating frequencies require 3-D electromagnetic methods,
such as the [[wikipedia:Partial_element_equivalent_circuit|Partial Element Equivalent Circuit]] (PEEC) method; it stems from the integral equation form of [[wikipedia:Maxwell's_equations|Maxwell's equations]].
The main difference of the PEEC method with other integral-Equation-based techniques, such as the method of moments, resides in the fact that it provides a circuit interpretation of the [[wikipedia:Electric-field_integral_equation|Electric Field Integral Equation]] (EFIE) in terms of partial elements, namely resistances, partial inductances, and coefficients of potential. In the standard approach, volumes and surfaces are discretized into elementary regions, hexahedra, and patches respectively over which the current and charge densities are expanded into a series of basis functions.
Pulse basis functions are usually adopted as expansion and weight functions.
Such choice of pulse basis functions corresponds to assuming constant current density and charge density over the elementary volume (inductive) and surface (capacitive) cells, respectively.
Following the standard Galerkin's testing procedure, topological elements, namely nodes and branches, are generated and electrical lumped elements are identified modeling both the magnetic and electric field coupling.
Conductors are modeled by their ohmic resistance, while dielectrics requires modeling the excess charge due to the dielectric polarization.
Magnetic and electric field coupling are modeled by partial inductances and coefficients of potential, respectively.

The magnetic field coupling between two inductive volume cells <math>\alpha</math> and <math>\beta</math> is described by the partial inductance

:<math> L_{p_{\alpha\beta}}=\frac{\mu}{4\pi}\frac{1}{a_{\alpha}a_{\beta}}\int_{u_{\alpha}}\int_{u_{\beta}}\frac{1}{R_{\alpha\beta}}\,du_{\alpha}\,du_{\beta} </math>

where <math>R_{\alpha\beta}</math> is the distance between any two points in the volumes <math>u_{\alpha}</math> and <math>u_{\beta}</math> with <math>a_{\alpha}</math> and <math>a_{\beta}</math> their cross section. The electric field coupling between two capacitive surface cells <math>\delta</math> and <math>\gamma</math> is modeled by the coefficient of the potential

:<math> P_{\delta\gamma}=\frac{1}{4\pi\epsilon}\frac{1}{S_{\delta}S_{\gamma}}\int_{S_{\delta}}\int_{S_{\gamma}}\frac{1}{R_{\delta\gamma}}\,dS_{\delta}\,dS_{\gamma} </math>

where <math>R_{\delta\gamma}</math> is the distance between any two points on the surfaces <math>\delta</math> and <math>\gamma</math>, while <math>S_{\delta}</math> and <math>S_{\gamma}</math> denote the area of their respective surfaces <ref name="ferranti11"/>.

Generalized Kirchhoff's laws for conductors, when dielectrics are considered, can be rewritten as

:<math> \textbf{P}^{-1}\frac{d\textbf{v}(t)}{dt}-\textbf{A}^T\textbf{i}(t)+\textbf{i}_e(t)=0, </math>

<figure id="fig:peec">[[File:Peec.jpg|400px|frame|<caption>Illustration of PEEC circuit electrical quantities for a conductor elementary cell (Figure from <ref name="ferranti11"/>).</caption>]]</figure>

:<math> -\textbf{A}\textbf{v}(t)-\textbf{L}_p\frac{d\textbf{i}(t)}{dt}-\textbf{v}_d(t)=0, </math>

:<math> \textbf{i}(t)=\textbf{C}_d\frac{d\textbf{v}_d(t)}{dt} </math>

where <math>\textbf{A}</math> is the connectivity matrix, <math>\textbf{v}(t)</math> denotes the node potentials to infinity, <math>\textbf{i}(t)</math> and <math>\textbf{i}_e(t)</math> represent the currents flowing in volume cells and the external currents, respectively, <math>\textbf{v}_d(t)</math> is the excess capacitance voltage drop, which is related to the excess charge by <math>\textbf{v}_d(t)=\textbf{C}_d^{-1}\textbf{q}_d(t)</math>. A selection matrix <math>\textbf{K}</math> is introduced to define the port voltages by selecting node potentials. The same matrix is used to obtain the external currents <math>\textbf{i}_e(t)</math> by the currents <math>\textbf{i}_s(t)</math>, which are of opposite sign with respect to the <math>n_p</math> port currents <math>\textbf{i}_p(t)</math>,

:<math> \textbf{v}_p(t)=\textbf{K}\textbf{v}(t), </math>

:<math> \textbf{i}_e(t)=\textbf{K}^T\textbf{i}_s(t). </math>

An example of PEEC circuit electrical quantities for a conductor elementary cell is illustrated, in the Laplace domain, in Fig. 1, where the current-controlled voltage sources <math>sL_{p,ij}I_j</math> and the current-controlled current sources <math>I_{cci}</math> model the magnetic and electric coupling, respectively.

Thus, assuming that we are interested in generating an admittance representation having <math>n_p</math> output currents under voltage excitation, and let us denote with <math>n_n</math> the number of nodes, <math>n_i</math> the number of branches where currents flow, <math>n_c</math> the number of branches of conductors, <math>n_d</math> the number of dielectrics, <math>n_d</math> the additional unknowns since dielectrics require the excess capacitance to model the polarization charge, and <math>n_u=n_i+n_d+n_n+n_p</math> the global number of unknowns, and if the [[wikipedia:Modified_nodal_analysis|Modified Nodal Analysis]] (MNA) approach is used, we have:

:::<math> \left[ \begin{array}{cccc} \textbf{P} & \textbf{0}_{n_n,n_i} & \textbf{0}_{n_n,n_d} & \textbf{0}_{n_n,n_p} \\ \textbf{0}_{n_i,n_n} & \textbf{L}_p & \textbf{0}_{n_i,n_d} & \textbf{0}_{n_i,n_p} \\ \textbf{0}_{n_d,n_n} & \textbf{0}_{n_d,n_i} & \textbf{C}_d & \textbf{0}_{n_d,n_p} \\ \textbf{0}_{n_p,n_n} & \textbf{0}_{n_p,n_i} & \textbf{0}_{n_p,n_d} & \textbf{0}_{n_p,n_p} \end{array}\right]\frac{d}{dt}\left[ \begin{array}{c}\textbf{q}(t) \\ \textbf{i}(t) \\ \textbf{v}_d(t) \\ \textbf{i}_s(t) \end{array}\right]= </math>

:<math>= - \left[ \begin{array}{cccc}\textbf{0}_{n_n,n_n} & -\textbf{P}\textbf{A}^T & \textbf{0}_{n_n,n_d} & \textbf{P}\textbf{K}^T \\ \textbf{AP} & \textbf{R} & \Phi & \textbf{0}_{n_i,n_p} \\ \textbf{0}_{n_d,n_n} & -\Phi^T & \textbf{0}_{n_d,n_d} & \textbf{0}_{n_d,n_p} \\ -\textbf{K}\textbf{P} & \textbf{0}_{n_p,n_i} & \textbf{0}_{n_p,n_d} & \textbf{0}_{n_p,n_p} \end{array}\right]\cdot\left[ \begin{array}{c} \textbf{q}(t) \\ \textbf{i}(t) \\ \textbf{v}_d(t) \\ \textbf{i}_s(t) \end{array}\right]+ \left[ \begin{array}{c}\textbf{0}_{n_n+n_i+n_d,n_p} \\ -\textbf{I}_{n_p,n_p} \end{array}\right] \cdot [ \textbf{v}_p(t) ]. </math>

Here <math>\textbf{0}</math> is a matrix of zeros, <math>\textbf{I}</math> is the identity matrix, both are with appropriate dimensions, and <math>\Phi=\left[ \begin{array}{c} \textbf{0}_{n_c,n_d} \\ \textbf{I}_{n_d,n_d} \end{array}\right]</math>. Then, in a more compact form, the above equation can be written as:

:<math> \left\{ \begin{array}{c} \textbf{C}\frac{d\textbf{x}(t)}{dt}=-\textbf{G}\textbf{x}(t)+\textbf{B}\textbf{u}(t)\\
\textbf{i}_p(t)=\textbf{L}^T\textbf{x}(t) \end{array}\right . \qquad (1)</math>

with <math>\textbf{x}(t)=\left[ \begin{array}{cccc} \textbf{q}(t)\quad\textbf{i}(t)\quad\textbf{v}_d(t)\quad\textbf{i}_s(t) \end{array}\right]^T</math>. Since this is an <math>n_p</math>-port formulation, whereby the only sources are the voltage sources at the <math>n_p</math>-ports nodes, <math>\textbf{B}=\textbf{L}</math> where <math>\textbf{B}\in\mathbb R^{n_u\times n_p}</math> (for more details on this model, refer to <ref name="ferranti11"/>).

==Motivation of MOR==

Since the number of equations produced by 3-D electromagnetic method PEEC is usually very large,
the inclusion of the PEEC model directly into a circuit simulator (like [[wikipedia:SPICE|SPICE]]) is computationally intractable for complex structures,
where the number of circuit elements can be tens of thousands.

==Data==

All data sets (in a MATLAB formatted data, downloadable in [[Media:TransmissionLines.rar|TransmissionLines.rar]]) in Fig. 1 are referred to as the multiconductor '''transmission lines''' in a MNA form, coming from the PEEC method (then, with dense matrices since they are obtained from the integral formulation of Maxwell's equation).
The LTI descriptor systems have the form of, equation <math>(1)</math>, where <math>C\in\mathbb R^{n\times n}</math> (with <math>C=C^T\ge0</math>), <math>G\in\mathbb R^{n\times n}</math> (with <math>G+G^T\ge0</math>), <math>B\in\mathbb R^{n\times m}</math> and <math>L=B^T</math>, <math>x(t)\in\mathbb R^n</math> is the vector of variables (charges, currents and node potential), the input signal <math>u(t)\in\mathbb R^m</math> are the sources (current or voltage generators depending on what one wants to analyze:
the impedances or the admittances) linked to some node, the output <math>y(t)\in\mathbb R^m</math> are the observation across the node where the sources are inserted. An accurate model of the dynamics of these data sets is generated between 10 kHz and 20 GHz.

<figtable id="tab:peec">
{| border="1"
! Name of the data set !! Matrices !! Dimension !! Number of inputs
|-
| <tt>dsysPEEC-MTLn1600m14</tt> || <math>G</math>, <math>B</math>, <math>C</math> (<math>L=B^T</math>) || 1600 || 14
|-
| <tt>dsysPEEC-MTLn2624m30</tt> || dss object (*) || 2624 || 30
|-
| <tt>dsysPEEC-MTLn5248m62</tt> || dss object (*) || 5248 || 62
|}
</figtable>

one can extract the matrices with Matlab command:

[G,B,L,D,C] = dssdata(dssObjectName);

e.g., if one wants to work on one of the last two data sets of this table,
just load it into the Matlab Workspace and type the command aforementioned on the Command Windows;
for the first example, once one loads the data, the Workspace shows directly the matrices.
Note that <math>D = 0</math>.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
E\dot{x}(t) &=& Ax(t) + Bu(t) \\
y(t) &=& C x(t)
\end{array}
</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}^{M \times N}</math>.

System variants:

<tt>dsysPEEC-MTLn1600m14</tt>: <math>N = 1600</math>, <math>M = 14</math>,
<tt>dsysPEEC-MTLn2624m30</tt>: <math>N = 2624</math>, <math>M = 30</math>,
<tt>dsysPEEC-MTLn5248m62</tt>: <math>N = 5248</math>, <math>M = 62</math>

==References==

<references>

<ref name="ferranti11"> F. Ferranti, G. Antonini, T. Dhaene, L. Knockaert, and A. E. Ruehli, "[https://doi.org/10.1109/TCPMT.2010.2101912 Physics-based passivity-preserving parameterized model order reduction for PEEC circuit analysis]", IEEE Transactions on Components, Packaging and Manufacturing Technology, 1(3): 399--409, 2011.</ref>

</references>

==Contact==

[[User:Deluca]]

[[User:Feng]]

Thermal Model

2023-11-30T09:42:48Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:first differential order]]
[[Category:time invariant]]
[[Category:MIMO]]
[[Category:Sparse]]

{{Infobox
|Title = Thermal Model
|Benchmark ID = thermalModel_n4257m1q7
|Category = oberwolfach
|System-Class = AP-LTI-FOS
|nstates = 4257
|ninputs = 1
|noutputs = 7
|nparameters = 3
|components = A, B, C, E
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
* [[User:Werner]]
|Zenodo-link = NA
}}

==Description: Boundary Condition Independent Thermal Model==
<figure id="fig1">[[File:ThermalModel.jpg|490px|thumb|right|<caption>A 2D-axisymmetrical model of the micro-thruster unit (not scaled). A heater is shown by a red spot.</caption>]]</figure>
A benchmark for the heat transfer problem with variable [[wikipedia:Heat_transfer_coefficient|film coefficients]] is presented.
It can be used to apply parametric model reduction algorithms to a linear first-order problem.

===Modeling===

An important requirements for a compact thermal model is that it should be independent from the boundary condition.
This means that a chip producer does not know conditions under which the chip will be used and hence the chip compact thermal model must allow an engineer to research on how the change in the environment influences the chip temperature.
The chip benchmarks representing boundary condition independent requirements are described in <ref name="lasance2001"/>.

Let us briefly describe the mathematical problem.
The thermal problem can be modeled by the heat transfer partial differential equation

:<math id="eq1">
\begin{align}
\nabla \cdot (\kappa(r)\nabla T(r,t)) + Q(r,t) - \rho(r)C_{p}(r)\frac{\partial T(r,t)}{\partial t} & = 0, & (1)
\end{align}
</math>

with <math>r</math> is the position, <math>t</math> is the time, <math>\kappa</math> is the thermal conductivity of the material, <math>C_{p}</math> is the specific heat capacity, <math>\rho</math> is the mass density, <math>Q</math> is the heat generation rate, and <math>T</math> is the unknown temperature distribution to be determined.
The heat exchange through device interfaces is usually modeled by convection boundary conditions

:<math id="eq2">
\begin{align}
q & = h_{i}(T - T_{bulk}), & (2)
\end{align}
</math>

where <math>q</math> is the heat flow through a given point, <math>h_{i}</math> is the film coefficient to describe the heat exchange for the <math>i</math>-th interface, <math>T</math> is the local temperature at this point, and <math>T_{bulk}</math> is the bulk temperature in the neighboring phase (in most cases <math>T_{bulk} = 0</math>).

After the discretization of equations [[#eq1|(1)]] and [[#eq2|(2)]] one obtains a system of ordinary differential equations as follows

:<math id="eq3">
\begin{align}
E \dot{x}(t) & = (A + \sum_{i} h_{i} A_{i})x(t) + B, & (3)
\end{align}
</math>

where <math>E</math> and <math>A</math> are the device system matrices, <math>A_i</math> is the diagonal matrix due to the discretization of equation [[#eq2|(2)]] for the <math>i</math>-th interface, <math>x(t)</math> is the vector with unknown temperatures.

In terms of the equation [[#eq3|(3)]] above, the engineering requirements read as follows.
A chip producer specifies the system matrices but the film coefficient, <math>h_i</math>, is controlled later on by another engineer.
As such, any reduced model to be useful should preserve <math>h_i</math> in the symbolic form.
This problem can be mathematically expressed as parametric model reduction<ref name="weile1999"/><ref name="gunupudi2003"/><ref name="daniel2004"/>.

Unfortunately, the benchmark from <ref name="lasance2001"/> is not available in the computer readable format.
For research purposes, we have modified a [[Micropyros_Thruster|Micropyros Thruster benchmark]] (see Fig. 1).
In the context of the present work, the model is as a generic example of a device with a single heat source when the generated heat dissipates through the device to the surroundings. The exchange between surrounding and the device is modeled by convection boundary conditions with different film coefficients at the top, <math>h_{top}</math>, bottom, <math>h_{bottom}</math>, and the side, <math>h_{side}</math>.
From this viewpoint, it is quite similar to a chip model used as a benchmark in <ref name="lasance2001"/>.
The goal of parametric model reduction in this case is to preserve <math>h_{top}</math>, <math>h_{bottom}</math>, and <math>h_{side}</math> in the reduced model in the symbolic form.

===Discretization===

We have used a 2D-axisymmetric microthruster model (T2DAL in [[Micropyros_Thruster|Micropyros Thruster]]).

The model has been made in [http://www.ansys.com/ ANSYS] and system matrices have been extracted by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem] <ref name="rudnyi2004"/>.
The benchmark contains a constant load vector.
The input function equal to one corresponds to the constant input power of <math>15 mW</math>.

The linear ordinary differential equations of first order are written as:

:<math>
\begin{align}
E \dot{x}(t) & = (A - h_{top} A_{top} - h_{bottom} A_{bottom} - h_{side} A_{side}) x(t) + B \\
y & = Cx(t)
\end{align}
</math>

where <math>E</math> and <math>A</math> are the symmetric sparse system matrices (heat capacity and heat conductivity matrix), <math>B</math> is the load vector, <math>C</math> is the output matrix, <math>A_{top}</math>, <math>A_{bottom}</math>, and <math>A_{side}</math> are the diagonal matrices from the discretization of the convection boundary conditions and <math>T</math> is the vector of unknown temperatures.

The numerical values of film coefficients, <math>h_{top}</math>, <math>h_{bottom}</math>, and <math>h_{side}</math> can be from <math>1</math> to <math>10^9</math>.
Typical important sets film coefficients can be found in <ref name="lasance2001"/>.
The allowable approximation error is <math>5\%</math> <ref name="lasance2001"/>.

The benchmark has been used in <ref name="feng2004"/><ref name="feng2005"/> where the problem is also described in more detail.

==Acknowledgements==

This work was partially funded by the DFG project '''MST-Compact (KO-1883/6)''', the Italian research council CNR together with the Italian province of Trento PAT, by the German Ministry of Research BMBF (SIMOD), and an operating grant of the University of Freiburg.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38865, see <ref name="rudnyi2005"/>.

==Data==

The matrices of this benchmark can be downloaded in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format:

* [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ThermalModel-dim1e3-T2DAL_BCI.tar.gz ThermalModel-dim1e3-T2DAL_BCI.tar.gz], 218.7 kB.

The matrix name is used as an extension of the matrix file. File <tt>T2DAL_BCI.C.names</tt> contains a list of output names written consecutively.

==Dimensions==

System structure:

:<math>
\begin{align}
E \dot{x}(t) & = (A_1 + h_2 A_2 + h_3 A_3 + h_4 A_4) x(t) + B \\
y(t) & = Cx(t)
\end{align}
</math>

System dimensions:

<math>E \in \mathbb{R}^{4257 \times 4257}</math>,
<math>A_1 \in \mathbb{R}^{4257 \times 4257}</math>,
<math>A_2 \in \mathbb{R}^{4257 \times 4257}</math>,
<math>A_3 \in \mathbb{R}^{4257 \times 4257}</math>,
<math>A_4 \in \mathbb{R}^{4257 \times 4257}</math>,
<math>B \in \mathbb{R}^{4257 \times 1}</math>,
<math>C \in \mathbb{R}^{7 \times 4257}</math>.

Parameter ranges:

<math>h_2 \in [1,10^9]</math>,
<math>h_3 \in [1,10^9]</math>,
<math>h_4 \in [1,10^9]</math>.

In the original Matrix Market data, <math>A_{top} = A_2, A_{bottom} = A_3, A_{side} = A_4</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection, '''Thermal Model'''. hosted at MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Thermal_Model

@MISC{morwiki_thermal,
author = <nowiki>{{Oberwolfach Benchmark Collection}}</nowiki>,
title = {Thermal Model},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Thermal_Model}</nowiki>,
year = 20XX
}

* For the background on the benchmark:

@TechReport{morFenRK04,
author = {Feng, L. and Rudnyi, E.~B. and Korvink, J.~G.},
title = {Parametric Model Reduction to Generate Boundary Condition Independent Compact Thermal Model},
institution = {IMTEK-Institute for Microsystem Technology},
type = {Technical report},
year = {2004},
url = <nowiki>{http://modelreduction.com/doc/papers/feng04THERMINIC.pdf}</nowiki>
}

==References==

<references>

<ref name="lasance2001">C.J.M. Lasance, "[https://doi.org/10.1109/6144.974943 Two benchmarks to facilitate the study of compact thermal modeling phenomena]", IEEE Transactions on Components and Packaging Technologies, 24: 559--565, 2001.</ref>

<ref name="weile1999">D.S. Weile, E. Michielssen, E. Grimme, K. Gallivan, "[https://doi.org/10.1016/S0893-9659(99)00063-4 A method for generating rational interpolant reduced order models of two-parameter linear systems]", Applied Mathematics Letters, 12: 93--102, 1999.</ref>

<ref name="gunupudi2003">P. K. Gunupudi, R. Khazaka, M. S. Nakhla, T. Smy, and D. Celo, "[https://doi.org/10.1109/TMTT.2003.820169 Passive parameterized time-domain macromodels for high-speed transmission-line networks]", IEEE Transactions on Microwave Theory and Techniques, 51: 2347--2354, 2003.</ref>

<ref name="daniel2004">L. Daniel, O.C. Siong, L.S. Chay, K.H. Lee, and J. White, "[https://doi.org/10.1109/TCAD.2004.826583 A Multiparameter Moment-Matching Model-Reduction Approach for Generating Geometrically Parameterized Interconnect Performance Models]", IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 23: 678--693, 2004.</ref>

<ref name="rudnyi2004">E.B. Rudnyi and J.G. Korvink, "[https://www.math.ucsd.edu/~helton/MTNSHISTORY/CONTENTS/2004LEUVEN/CDROM/papers/513.pdf Model Order Reduction of MEMS for Efficient Computer Aided Design and System Simulation]", MTNS2004, Sixteenth International Symposium on Mathematical Theory of Networks and Systems, Katholieke Universiteit Leuven, Belgium, July 5-9, 2004.</ref>

<ref name="feng2004">L. Feng, E.B. Rudnyi, J.G. Korvink, "[http://modelreduction.com/doc/papers/feng04THERMINIC.pdf Parametric Model Reduction to Generate Boundary Condition Independent Compact Thermal Model]", THERMINIC 2004, 10th International Workshop on Thermal Investigations of ICs and Systems, 29 September - 1 October 2004, Sophia Antipolis, Cote d'Azur, France.</ref>

<ref name="feng2005">L. Feng, E. B. Rudnyi, J. G. Korvink, "[https://doi.org/10.1109/TCAD.2005.852660 Preserving the film coefficient as a parameter in the compact thermal model for fast electro-thermal simulation]", IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 24(12): 1838--1847, 2005.</ref>

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

<ref name="rudnyi2005">E.B. Rudnyi, J.G. Korvink, [https://doi.org/10.1007/3-540-27909-1_17 Boundary Condition Independent Thermal Model], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 345--348, 2005.</ref>

</references>

==Contact==

'' [[User:Feng|Lihong Feng]] ''

Thermal Block

2023-11-30T09:42:29Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:first differential order]]
[[Category:time invariant]]
[[Category:MIMO]]
[[Category:Sparse]]

{{Infobox
|Title = Thermal Block
|Benchmark ID = thermalBlock_n7488m1q4
|Category = misc
|System-Class = AP-LTI-FOS
|nstates = 7488
|ninputs = 1
|noutputs = 4
|nparameters = 4
|components = A, B, C, E
|License = BSD 2-Clause "Simplified" License
|Creator = [[User:Saak]]
|Editor =
* [[User:Saak]]
* [[User:Himpe]]
* [[User:Mlinaric]]
|Zenodo-link = https://zenodo.org/record/3691894/files/ABCE.mat
}}

==Description==
A parametric semi-discretized heat transfer problem with varying heat transfer coefficients, the parameters, on subdomains. This model is also called the ''cookie baking problem'', and can be viewed as a flattened 2-D version of the ''skyscraper problem'' from high-performance computing.

<figure id="fig1">[[File:ThermalBlockDomain.svg|490px|thumb|right|<caption>The computational domain and boundaries.</caption>]]</figure>
<figure id="fig1">[[File:ThermalBlockTend.png|490px|thumb|right|<caption>A sample heat distribution at time 1.0 for parameter choice [100, 0.01, 0.001, 0.0001].</caption>]]</figure>
<figure id="fig1">[[File:ThermalBlockSigmaMagnitude.png|490px|thumb|right|<caption>Sigma magnitude plot of the single parameter variant.</caption>]]</figure>

===Modeling===
Consider a parameter <math>\mu\in{[10^{-6},10^2]}^4\subset\mathbb{R}^{4}</math> and define the heat conductivity <math>\sigma(\xi; \mu)</math> as <math>1.0</math> when <math>\xi\in\Omega_0</math> and <math>\sigma(\xi; \mu)=\mu_i</math> when <math>\xi\in\Omega_i</math>. The heat distribution is governed by the equation:
:<math>
\partial_t \theta(t, \xi; \mu) + \nabla \cdot (- \sigma(\xi; \mu) \nabla \theta(t, \xi; \mu)) = 0,\text{ for } t\in (0,T), \text{ and } \xi \in \Omega,
</math>
with a heat-inflow condition on the left ([[wikipedia:Neumann_boundary_condition|Neumann boundary]])
:<math>
\sigma(\xi; \mu) \nabla \theta(t, \xi; \mu) \cdot n(\xi) = u(t)\text{ for } t \in (0,T), \text{ and } \xi \in \Gamma_{in},
</math>
perfect isolation on the top and bottom ([[wikipedia:Boundary_conditions_in_fluid_dynamics#Wall_boundary_condition|Neumann-zero boundary]])
:<math>
\sigma(\xi; \mu) \nabla \theta(t, \xi; \mu) \cdot n(\xi) = 0\text{ for } t \in (0,T), \text{ and } \xi \in \Gamma_N,
</math>
and fixed temperature on the right ([[wikipedia:Dirichlet_boundary_condition|Dirichlet boundary]])
:<math>
\theta(t, \xi; \mu) = 0\text{ for } t \in (0,T), \text{ and } \xi \in \Gamma_{D},
</math>
and initial condition
:<math>
\theta(0, \xi; \mu) = 0 \text{ for } \xi \in \Omega.
</math>

===Discretization===
For the discretization, [https://fenicsproject.org/ FEniCS] '''2019.1''' was used on a simplicial grid with first order elements. The mesh is generated from the domain specification using [http://gmsh.info/ gmsh] '''3.0.6''' with '<code>clscale</code>' set to <math>0.1</math>. The Python-based source code for the discretization can be found at [https://doi.org/10.5281/zenodo.3691894 Zenodo].

==Origin==
This benchmark was developed for the [https://imsc.uni-graz.at/modred2019/ MODRED 2019] proceedings<ref name="morRavS20/>.

==Data==
The benchmark includes the basic domain description as a gmsh input file, Python scripts for the matrix assembly, simulation in pyMOR, and visualization as VTK, together with the matrices both as one combined file <code>ABCE.mat</code> or separate matrix market files for all matrices. The sources and the <code>ABCE.mat</code> are available for download at [https://doi.org/10.5281/zenodo.3691894 Zenodo].

Note that the heat transfer coefficients are designed as characteristic functions on the domains, such that the system is only well-posed when all entries in <math>\mu</math> are positive.

==Dimensions==
System structure:
:<math>
\begin{align}
E\dot{x}(t) &= (A_1 + \mu_1 A_2 + \mu_2 A_3 + \mu_3 A_4 + \mu_4 A_5) x(t) + Bu(t), \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>E \in \mathbb{R}^{N \times N}</math>,
<math>A_{1,...,5} \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times 1}</math>,
<math>C \in \mathbb{R}^{4 \times N}</math>,

where <math>N=7\,488</math> for the system matrices given in <code>ABCE.mat</code>.

==Variants==
Besides the full four parameter setup, the model can be used in variations with other numbers of independent parameters. The following two are recommended in the original work and have been investigated in the literature<ref name="morBenW20c"/>,<ref name="morHim20"/>,<ref name="morBenKS20"/>,<ref name="morMliRS20"/>.

===Single parameter===
The interpretation of the thermal block as the "cookie baking" problem with slight variation in the dough leads to an easy one parameter variant. Here the new single parameter <math>\hat\mu\in [ 10^{-6}, 10^2]</math> is chosen such that <math> \mu = \hat\mu\left[0.2, 0.4, 0.6, 0.8\right]. </math>

===Non-parametric===
The system can be used as a standard LTI state-space model. It is suggested to use <math>\mu = \sqrt{10} [0.2, 0.4, 0.6, 0.8]</math>.

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

* For the benchmark itself and its data:
:: S. Rave and J. Saak, '''Thermal Block'''. MORwiki - Model Order Reduction Wiki, 2020. http://modelreduction.org/index.php/Thermal_Block

@MISC{morwiki_thermalblock,
author = <nowiki>{Rave, S. and Saak, J.}</nowiki>,
title = {Thermal Block},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Thermal_Block}</nowiki>,
year = 2020
}

* For the background on the benchmark:
:: S. Rave and J. Saak, [https://arxiv.org/abs/2003.00846 '''A Non-Stationary Thermal-Block Benchmark Model for Parametric Model Order Reduction''']. e-prints 2003.00846, arXiv, math.NA (2020).

@INPROCEEDINGS{morRavS21,
author = {Rave, S. and Saak, J.},
title = {A Non-Stationary Thermal-Block Benchmark Model for Parametric Model Order Reduction},
booktitle = {Model Reduction of Complex Dynamical Systems},
series = {International Series of Numerical Mathematics},
volume = {171},
publisher = {Springer},
year = 2021,
doi = {10.1007/978-3-030-72983-7_16}
}

==References==
<references>
<ref name="morBenW20c">P. Benner, S. W. R. Werner, [https://doi.org/10.1007/978-3-030-72983-7_19 MORLAB -- the Model Order Reduction LABoratory],
Model Reduction of Complex Dynamical Systems, International Series of Numerical Mathematics 171: 393--415, 2021.</ref>
<ref name="morHim20">C. Himpe, [https://doi.org/10.1007/978-3-030-72983-7_7 Comparing (empirical-Gramian-based) model order reduction algorithms],
Model Reduction of Complex Dynamical Systems, International Series of Numerical Mathematics 171: 141--164, 2021.</ref>
<ref name="morBenKS20">P. Benner, M. Köhler, J. Saak, [https://doi.org/10.1007/978-3-030-72983-7_18 Matrix equations, sparse solvers: M-M.E.S.S.-2.0.1 – philosophy, features and application for (parametric) model order reduction], Model Reduction of Complex Dynamical Systems, International Series of Numerical Mathematics 171: 369--392, 2021.</ref>
<ref name="morRavS20">S. Rave, J. Saak, [https://doi.org/10.1007/978-3-030-72983-7_16 An Non-Stationary Thermal-Block Benchmark Model for Parametric Model Order Reduction], Model Reduction of Complex Dynamical Systems, International Series of Numerical Mathematics 171: 349-356, 2021.</ref>
<ref name="morMliRS20">P. Mlinarić, S. Rave, J. Saak, [https://doi.org/10.1007/978-3-030-72983-7_17 Parametric model order reduction using pyMOR], Model Reduction of Complex Dynamical Systems, International Series of Numerical Mathematics 171: 357--367 , 2021.</ref>
</references>

==Contact==
'' [[User:Saak]]''

Supersonic Engine Inlet

2023-11-30T09:42:04Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:Differential_algebraic_system]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:MIMO]]
[[Category:Sparse]]

{{Infobox
|Title = Supersonic Engine Inlet
|Benchmark ID = supersonicEngineInlet_n11730m2q1
|Category = oberwolfach
|System-Class = LTI-FOS
|nstates = 11730
|ninputs = 2
|noutputs = 1
|nparameters = 0
|components = A, B, C, E
|License = NA
|Creator = [[User:Werner]]
|Editor =
* [[User:Werner]]
* [[User:Himpe]]
|Zenodo-link = NA
}}

==Description: Active Control of a Supersonic Engine Inlet==

<figure id="fig1">[[File:Supersonic_Inlet1.png|750px|thumb|right|<caption>Steady-state Mach contours inside diffuser. Freestream Mach number
is 2.2.</caption>]]</figure>

This example considers unsteady flow through a [[wikipedia:Diffuser_(thermodynamics)#Supersonic_Diffusers|supersonic diffuser]] as shown in Fig. 1.
The diffuser operates at a nominal [[wikipedia:Mach_number|Mach number]] of <math>2.2</math>, however it is subject to perturbations in the incoming flow, which may be due (for
example) to atmospheric variations.
In nominal operation, there is a strong shock downstream of the diffuser throat, as can be seen from the Mach contours
plotted in Figure Fig. 1.
Incoming disturbances can cause the shock to move forward towards the throat. When the shock sits at the throat, the inlet
is unstable, since any disturbance that moves the shock slightly upstream will cause it to move forward rapidly, leading to unstart of the inlet. This is extremely undesirable, since unstart results in a large loss of thrust.
In order to prevent unstart from occurring, one option is to actively control the position of the shock.
This control may be effected through flow bleeding upstream of the diffuser throat.

A complete description of the benchmark and some model reduction results can be downloaded as PDF file [https://www.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SupersonicEngineInlet.pdf here].

===Active Flow Control Setup===

<figure id="fig2">[[File:Supersonic_Inlet2.png|600px|thumb|right|<caption>Supersonic diffuser active flow control problem setup.</caption>]]</figure>

Fig. 2 presents the schematic of the actuation mechanism.
Incoming flow with possible disturbances enters the inlet and is sensed using pressure sensors.
The controller then adjusts the bleed upstream of the throat in order to control the position of the shock and to prevent it from moving upstream.
In simulations, it is difficult to automatically determine the shock location.
The average Mach number at the diffuser throat provides an appropriate surrogate that can be easily computed.
There are several transfer functions of interest in this problem.
The shock position will be controlled by monitoring the average Mach number at the diffuser throat.
The reduced-order model must capture the dynamics of this output in response to two inputs: the incoming flow disturbance and the bleed actuation.
In addition, total pressure measurements at the diffuser wall are used for sensing.

===CFD Formulation===

The unsteady, two-dimensional flow of an inviscid, compressible fluid is governed by the Euler equations.
The usual statements of mass, momentum, and energy can be written in integral form as

:<math>
\begin{align}
\frac{\partial}{\partial t}\iint\rho\mathrm{d}V + \oint\rho Q\cdot\mathrm{dA} & = 0,\\
\frac{\partial}{\partial t}\iint\rho Q\mathrm{d}V + \oint\rho Q (Q\cdot\mathrm{dA}) + \oint p \mathrm{dA} & = 0,\\
\frac{\partial}{\partial t}\iint\rho E\mathrm{d}V + \oint\rho H (Q\cdot\mathrm{dA}) + \oint p Q\cdot\mathrm{dA} & = 0,
\end{align}
</math>

where <math>\rho</math>, <math>Q</math>, <math>H</math>, <math>E</math>, and <math>p</math> denote density, flow velocity, total enthalpy, energy, and pressure, respectively.
The CFD formulation for this problem uses a finite volume method and is described fully in <ref name="lassaux2002"/> and <ref name="willcox2005"/>.
The unknown flow quantities used are the density, streamwise velocity component, normal velocity component, and enthalpy at each point in the computational grid.
Note that the local flow velocity components <math>q</math> and <math>q^{\perp}</math> are defined using a streamline computational grid that is computed for the steady-state solution.
<math>q</math> is the projection of the flow velocity on the meanline direction of the grid cell, and <math>q^{\perp}</math> is the normal-to-meanline component.
To simplify the implementation of the integral energy equation, total enthalpy is also used in place of energy.
The vector of unknowns at each node <math>i</math> is therefore

:<math>
x_{i} = \begin{bmatrix} \rho_{i}, & q_{i}, & q^{\perp}_{i}, & H_{i} \end{bmatrix}^{T}.
</math>

Two physically different kinds of boundary conditions exist: inflow/outflow conditions, and conditions applied at a solid wall.
At a solid wall, the usual no-slip condition of zero normal flow velocity is easily applied as <math>q^{\perp} = 0</math>.
In addition, we will allow for mass addition or removal (bleed) at various positions along the wall.
The bleed condition is also easily specified.
We set

:<math>
q^{\perp} = \frac{\dot{m}}{\rho},
</math>

where <math>\dot{m}</math> is the specified mass flux per unit length along the bleed slot.
At inflow boundaries, Riemann boundary conditions are used.
For the diffuser problem considered here, all inflow boundaries are supersonic, and hence we impose inlet vorticity, entropy and Riemann’s invariants.
At the exit of the duct, we impose outlet pressure.

===Linearized CFD Matrices===

The two-dimensional integral Euler equations are linearized about the steady-state solution to obtain an unsteady system of the form

:<math>
\begin{align}
E\dot{x}(t) & = Ax(t) + Bu(t),\\
y(t) & = Cx(t).
\end{align}
</math>

The descriptor matrix <math>E</math> arises from the particular CFD formulation.
In addition, the matrix <math>E</math> contains some zero rows that are due to implementation of boundary conditions.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38866, see also <ref name="willcox2005"/>.

==Data==

The matrices of this benchmark can be downloaded in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format:
* [http://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SupersonicEngineInlet-dim1e4-Inlet.tar.gz SupersonicEngineInlet-dim1e4-Inlet.tar.gz].
The size of the file is 5.4 MB.
The matrix name is used as an extension of the matrix file.

==Dimensions==

System structure:

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

System dimensions:

<math>E \in \mathbb{R}^{11730 \times 11730}</math>,
<math>A \in \mathbb{R}^{11730 \times 11730}</math>,
<math>B \in \mathbb{R}^{11730 \times 2}</math>,
<math>C \in \mathbb{R}^{1 \times 11730}</math>

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection, '''Supersonic Engine Inlet'''. hosted at MORwiki - Model Order Reduction Wiki, 2005. http://modelreduction.org/index.php/Supersonic_Engine_Inlet

@MISC{morwiki_supsonengine,
author = <nowiki>{{Oberwolfach Benchmark Collection}}</nowiki>,
title = {Supersonic Engine Inlet},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Supersonic_Engine_Inlet}</nowiki>,
year = 2005
}

* For the background on the benchmark:

@MASTERSTHESIS{morLas13,
author = {G. Lassaux},
year = 2002,
title = {High-Fidelity Reduced-Order Aerodynamic Models: Application to
Active Control of Engine Inlets},
school = {Massachusetts Institute of Technology},
address = {Cambridge, USA},
url = <nowiki>{http://web.mit.edu/kwillcox/Public/Web/LassauxMS.pdf}</nowiki>
}

==References==

<references>

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

<ref name="lassaux2002">G. Lassaux. [http://hdl.handle.net/1721.1/82238 High-Fidelity Reduced-Order Aerodynamic Models: Application to Active Control of Engine Inlets]. Master’s thesis, Dept. of Aeronautics and Astronautics, MIT, June 2002.</ref>

<ref name="willcox2005">K. Willcox , G. Lassaux, [https://doi.org/10.1007/3-540-27909-1_20 Model Reduction of an Actively Controlled Supersonic Diffuser]. In: Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 357--361, 2005.</ref>

</references>

Sound transmission through a plate

2023-11-30T09:41:24Z

Malhotra: infobox string edit

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

{{Infobox
|Title = Sound Transmission through a Plate
|Benchmark ID = soundTransmission_n95480m1q1
|Category = misc
|System-Class = LTI-SOS
|nstates = 95480
|ninputs = 1
|noutputs = 1
|nparameters = 0
|components = B, C, E, K, M
|License = Creative Commons Attribution 4.0 International
|Creator = [[User:Aumann]]
|Editor =
* [[User:Aumann]]
|Zenodo-link = https://zenodo.org/record/7670587/files/soundTransmission_n95480m1q1.mat
}}

<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>

Sound transmission through a plate

2023-11-30T09:40:56Z

Malhotra: infobox string

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

{{Infobox
|Title = Silicon Nitride Membrane
|Benchmark ID = siliconNitrideMembrane_n60020m1q2
|Category = misc
|System-Class = AP-LTI-FOS
|nstates = 60020
|ninputs = 1
|noutputs = 2
|nparameters = 2
|components = A, B, C, E
|License = NA
|Creator = [[User:Feng]]
|Editor =
* [[User:Feng]]
* [[User:Himpe]]
|Zenodo-link = NA
}}

<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>

Silicon Nitride Membrane

2023-11-30T09:40:37Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Parametric]]
[[Category:linear]]
[[Category:first differential order]]
[[Category:time invariant]]

{{Infobox
|Title = Silicon Nitride Membrane
|Benchmark ID = siliconNitrideMembrane_n60020m1q2
|Category = misc
|System-Class = AP-LTI-FOS
|nstates = 60020
|ninputs = 1
|noutputs = 2
|nparameters = 2
|components = A, B, C, E
|License = NA
|Creator = [[User:Feng]]
|Editor =
* [[User:Feng]]
* [[User:Himpe]]
|Zenodo-link = NA
}}

==Description==

<figure id="fig:tempprof">[[File:Fig_1.png|right|frame|<caption>silicon nitride membrane temperature profile</caption>]]</figure>

A '''silicon nitride membrane''' ([[wikipedia:Silicon_nitride|SiN]] membrane) <ref name="bechthold10"/> can be a part of a gas sensor,
but also a part of an infra-red sensor, a microthruster, an optical filter etc.
This structure resembles a microhotplate similar to other micro-fabricated devices such as gas sensors <ref name="spannhake05"/> and infrared sources <ref name="graf04"/> (See also [[Gas_Sensor|Gas Sensor Benchmark]]).
See Fig. 1, the temperature profile for the SiN membrane.

The governing heat transfer equation in the membrane is:

:<math> \nabla \cdot (\kappa \nabla T)+Q - \rho c_p \cdot \frac{\partial T}{\partial t}=0 </math>

where <math>\kappa </math> is the thermal conductivity in <math>W m^{-1} K^{-1}</math>,
<math>c_p</math> is the specific heat capacity in <math>J kg^{-1} K^{-1}</math>,
<math>\rho</math> is the mass density in <math>kg m^{-3}</math> and <math>T</math> is the temperature distribution.
We assume a homogeneous heat generation rate over a lumped resistor:

:<math>Q = \frac{u^2(t)}{R(T)}</math>

with <math>Q</math> the heat generation rate per unit volume in <math>W m^{-3}</math>.
We use the initial condition <math>T_0 = 273K</math>,
and the Dirichlet boundary condition <math>T = 273 K</math> at the bottom of the computational domain.

The convection boundary condition at the top of the membrane is

:<math>
q=h(T-T_{air}),
</math>

where <math>h</math> is the heat transfer coefficient between the membrane and the ambient air in <math>W m^{-2} K^{-1}</math>.

==Discretization==

Under the above convection boundary condition and assuming <math>T_{air}=0</math>,
a finite element discretization of the heat transfer model leads to the parametrized system as below,

:<math>
(E_0+\rho c_p \cdot E_1) \dot{T} +(A_0 +\kappa \cdot A_1 +h \cdot A_2)T = B \frac{u^2(t)}{R(T)}, \quad
y=C T,
</math>

where the volumetric heat capacity <math>\rho c_p</math>,
thermal conductivity <math>\kappa</math> and the heat transfer coefficient <math>h</math> between the membrane are kept as parameters.
The volumetric hear capacity <math>\rho c_p</math> is the product of two independent variables,
i.e. the specific hear capacity <math>c_p</math> and the density <math>\rho</math>.
The range of interest for the four independent variables are <math>\kappa \in [2, 5]</math>, <math>c_p \in [400, 750]</math>, <math>\rho \in [3000,3200]</math> and <math> h \in [10, 12]</math> respectively.
The frequency range is <math>f \in [0,25]Hz</math>.
What is of interest is the output in time domain.
The interesting time interval is <math>t \in [0,0.04]s</math>.

Here <math>R(T)</math> is either a constant heat resistivity <math>R(T)=R_0</math>, or <math>R(T)=R_0(1+\alpha T)</math>,
which depends linearly on the temperature.
Here we use <math>R_0=274.94 \Omega</math> and temperature coefficient <math>\alpha=2.293 \pm 0.006 \times 10^{-4}</math>.
The model was created and meshed in ANSYS. It contains a constant load vector <math>B</math> corresponding to the constant input power of <math>2.49mW</math>.
The number of degrees of freedom is <math>n=60,020</math>.

The input function <math>u(t)</math> is a step function with the value <math>1</math>,
which disappears at the time <math>0.02s</math>.
This means between <math>0s</math> and <math>0.02s</math> input is one and after that it is zero.
However, be aware that <math>u(t)</math> is just a factor with which the load vector B is multiplied and which corresponds to the heating power of <math>2.49mW</math>.
This means if one keeps <math>u(t)</math> as suggested above, the device is heated with <math>2.49mW</math> for the time length of 0.02s and after that the heating is turned off.
If for whatever reason, one wants the heating power to be <math>5mW</math>, then <math>u(t)</math> has to be set equal to two, etc.
When <math>R(T)=R_0(1+\alpha T)</math>, it is a function of the state vector <math>T</math> and hence, the system has non-linear input (It is also called a weakly nonlinear system.).

==Data==

The model is generated in ANSYS. The system matrices are in [http://math.nist.gov/MatrixMarket/ MatrixMarket] format and can be downloaded here: [[Media: SiN_membrane.tgz|SiN_membrane.tgz]].

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
(E_1 + \rho c_p E_2)\dot{x}(t) &=& -(A_1 + \kappa A_2 + h A_3)x(t) + Bu(t) \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>E_{1,2} \in \mathbb{R}^{60020 \times 60020}</math>,
<math>A_{1,2,3} \in \mathbb{R}^{60020 \times 60020}</math>,
<math>B \in \mathbb{R}^{60020 \times 1}</math>,
<math>C \in \mathbb{R}^{2 \times 60020}</math>.

==References==

<references>

<ref name="bechthold10">T. Bechtold, D. Hohfeld, E. B. Rudnyi and M. Guenther, "[https://doi.org/10.1088/0960-1317/20/4/045030 Efficient extraction of thin-film thermal parameters from numerical models via parametric model order reduction]", J. Micromech. Microeng. 20(4): 045030, 2010.</ref>

<ref name="spannhake05">J. Spannhake, O. Schulz, A. Helwig, G. Müller and T. Doll, "[https://doi.org/10.1109/ICSENS.2005.1597811 Design, development and operational concept of an advanced MEMS IR source for miniaturized gas sensor systems]", Proc. Sensors: 762--765, 2005.</ref>

<ref name="graf04">M. Graf, D. Barrettino, S. Taschini, C. Hagleitner, A. Hierlemann and H. Baltes, "[https://doi.org/10.1021/ac035432h Metal oxide-based monolithic complementary metal oxide semiconductor gas sensor microsystem]", Anal. Chem., 76(15): 4437--4445, 2004.</ref>

</references>

==Contact==
'' [[User:Feng|Lihong Feng]] ''

[http://www.igs.uni-rostock.de/mitarbeiter/tamara-bechtold/ Tamara Bechtold]

Scanning Electrochemical Microscopy

2023-11-30T09:40:12Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:parametric]]
[[Category:linear]]
[[Category:time varying]]
[[Category:Parametric]]
[[Category:first differential order]]
[[Category:nonzero initial condition]]

{{Infobox
|Title = Scanning Electrochemical Microscopy
|Benchmark ID = scanningElectrochemicalMicroscopy_n16912m1q5
|Category = misc
|System-Class = AP-LTI-FOS
|nstates = 16912
|ninputs = 1
|noutputs = 5
|nparameters = 2
|components = A, B, C, E
|License = NA
|Creator = [[User:Feng]]
|Editor =
* [[User:Feng]]
* [[User:Himpe]]
* [[User:Will]]
* [[User:Lund]]
|Zenodo-link = NA
}}

==Description==

<figure id="fig:cylin">
[[Image:Fig.1.JPG|thumb|right|300px|<caption>Cylindrical Electrode</caption>]]
</figure>

'''Scanning Electrochemical Microscopy''' (SECM) has many applications in current problems in the biological field.
Quantitative mathematical models have been developed for different operating modes of the SECM.
Except for some very specific problems, like the diffusion-controlled current on a circular electrode far away from the border,
solutions can only be obtained by numerical simulation, which is based on discretization of the model in space by an appropriate method like finite differences, finite elements, or boundary elements.
After discretization, a high-dimensional system of ordinary differential equations is obtained.
Its high dimensionality leads to high computational cost.

We consider a cylindrical electrode in Fig. 1.
The computation domain under the 2D-axisymmetric approximation includes the electrolyte under the electrode.
We assume that the concentration does not depend on the rotation angle.
A single chemical reaction takes place on the electrode:

:<math>Ox+e^-\Leftrightarrow Red, \quad \quad \quad \quad (1) </math>

where <math>Ox</math> and <math>Red</math> are two different species in the reaction.
According to the theory of SECM <ref name="mirkin01"/>, the species transport in the electrolyte is described by diffusion only.
The diffusion partial differential equation is given by the second [[wikipedia:Fick's_laws_of_diffusion|Fick's law]] as follows

:<math>
\frac{dc_1}{dt}=D_1\cdot \Delta ^2c_1 , \quad
\frac{dc_2}{dt}=D_2\cdot \Delta ^2c_2,
</math>

where <math>c_1</math> and <math>c_2</math> are the concentration fields of species <math>Ox</math> and <math>Red</math>, respectively.
The initial conditions are <math>c_1(0)=c_{1,0}</math> and <math>c_2(0)=c_{2,0}.</math> conditions at the glass and the bottom of the bath
are described by the Neumann boundary conditions of zero flux <math>\nabla c_1\cdot \vec{n}=0</math> and <math>\nabla c_2\cdot \vec{n}=0</math>.
Conditions at the border of the bulk are described by Dirichlet boundary conditions of constant concentration, equal to the initial conditions <math>c_1=c_{1,0}</math> and <math>c_2=c_{2,0}</math>.
The boundary conditions at the electrode are described by

:<math>
\nabla c_1\cdot \vec{n}=j, \,
\nabla c_2\cdot \vec{n}=-j. \quad \quad \quad \quad (2)
</math>

Here <math>j</math> is related to the forward reaction rate <math>k_f</math> and the backward reaction rate <math>k_b</math> through the [[wikipedia:Butler–Volmer_equation|Butler-Volmer equation]],

:<math>
j=k_f \cdot c_1-k_b \cdot c_2.
</math>

The reaction rates <math>k_f</math> and <math>k_b</math> are in the following form,

:<math>
k_f=k^0\exp{(\frac{\alpha z F(v(t)-v^0)}{RT})}, </math>

:<math>k_b=k^0\exp{(\frac{-(1-\alpha) z F(v(t)-v^0)}{RT})} .
</math>

Here, <math>k^0</math> is the heterogeneous standard rate constant, which is an empirical transmission factor for a heterogeneous reaction.
<math>F</math> is the [[wikipedia:Faraday_constant|Faraday-constant]], <math>R</math> is the [[wikipedia:Gas_constant|gas constant]], <math>T</math> is the temperature, and <math>z</math> is the number of exchanged electrons per reaction.
<math>u(t)=v(t)-v^0</math> is the difference between the electrode potential and the reference potential.
This difference, to which we refer below as voltage, changes during the measurement of a [[wikipedia:Voltammetry|voltammogram]].

==Model==

The control volume method has been used for the spatial discretization of (1).
Together with the boundary conditions, the resulting system of ordinary differential equations is as follows,

:<math>
E\frac{d\vec{c}}{dt}+K(u(t))\vec{c}-A\vec{c}=B,\quad
y(t)=C\vec{c},\quad
\vec{c}(0)=\vec{c}_0 \neq 0,
</math>

where E and <math>K(u(t))</math> are system matrices, <math>K(u(t))</math> is a function of voltage that in turn depends on time.
The voltage appears in the system matrix due to the boundary conditions (2).
The vector <math>\vec{c} \in \mathbb{R}^n</math> is the vector of unknown concentrations, which includes both the <math>Ox</math> and <math>Red</math> species.
The vector <math>B</math> is the load vector, which arises as a consequence of the Dirichlet boundary conditions imposed at the bulk boundary of the electrolyte. The total current is computed as an integral (sum) over the electrode surface.
The matrix <math>K(u(t))</math> has the following form,

:<math>
K(u(t))=K_1(u(t))+K_2(u(t)),
</math>

where <math>K_i(u(t))=h_i D_i, \, i=1,2,</math> and <math>h_1=\exp(\beta u(t)), \, h_2=\exp(-\beta u(t))</math>.
The voltage <math>u(t)</math> is a function of <math>\sigma</math>,

:<math>
u(t)=\sigma t-1, \, \text{for } t \leq \frac{2}{ \sigma}, \quad
u(t)=-\sigma t+3, \, \text{for } \frac{2}{ \sigma} < t \leq \frac{4}{ \sigma},
</math>

where <math>\sigma</math> can take four different values, <math>\sigma=0.5, \, 0.05, \, 0.005, \, 0.0005</math>. The constant <math>\beta</math> is computed from the parameters <math>\alpha, \, z, \, F, \, R,</math> and <math>T,</math> leading to the value <math>\beta=21.243036728240824</math>.
Although the system is a time-varying system, it can be considered as a parametrized system with two parameters <math>h_1</math> and <math>h_2</math>.

==Data==

The data of the system matrices <math>E, \ D_1, \ D_2, \ A, \ B, C</math> as well as the initial state <math>\vec{c}_0=x_0</math> are in MatrixMarket format (http://math.nist.gov/MatrixMarket/), and can be downloaded here [[Media:SECM.TGZ|SECM.tgz]]. The quantity of interest is the current which is computed by <math>I(t)=C(5,:)\vec{c}</math> in MATLAB notation. The associated plot is called the [[wikipedia:Cyclic_voltammetry|cyclic voltammogram]] <ref name="feng06"/>, which is the plot of the current changing with the voltage <math>u(t)</math>.

For the MOR Benchmark tool ([[MORB]]), the matrices <math>A, D_1, D_2</math> have been renamed <math>A_1, A_2, A_3</math>, respectively.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
E\dot{c}(t) &=& (A_1 - h_1 A_2 - h_2 A_3)c(t) + Bu(t) \\
y(t) &=& Cc(t)
\end{array}
</math>

System dimensions:

<math>E \in \mathbb{R}^{16\,912 \times 16\,912}</math>,
<math>A_{1,2,3} \in \mathbb{R}^{16\,912 \times 16\,912}</math>,
<math>B \in \mathbb{R}^{16\,912 \times 1}</math>,
<math>C \in \mathbb{R}^{5 \times 16\,912}</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Scanning Electrochemical Microscopy'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Scanning_Electrochemical_Microscopy

@MISC{morwiki_secm,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Scanning Electrochemical Microscopy},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Scanning_Electrochemical_Microscopy}</nowiki>,
year = {2018}
}

* For the background on the benchmark:

@ARTICLE{morFenKRetal06,
author = {L. Feng, D. Koziol, E.B. Rudnyi, and J.G. Korvink},
title = {Parametric Model Reduction for Fast Simulation of Cyclic Voltammograms},
journal = {Sensor Letters},
volume = 4,
number = 2,
pages = {165--173},
year = 2006,
doi = {10.1166/sl.2006.021}
}

==References==

<references>

<ref name="mirkin01"> M.V. Mirkin, "[https://doi.org/10.1201/b11850-7 Chapter 5: Theory]", In: A.J. Bard and M.V. Mirkin, (eds.), Scanning Electrochemical Microscopy, CRC Press: 144--199, 2001.</ref>

<ref name="feng06"> L. Feng, D. Koziol, E.B. Rudnyi, and J.G. Korvink, "[https://doi.org/10.1166/sl.2006.021 Parametric Model Reduction for Fast Simulation of Cyclic Voltammograms]", Sensor Letters, 4(2): 165--173, 2006.</ref>

</references>

==Contact==

'' [[User:Feng|Lihong Feng]] ''

RCL Circuit Equations

2023-11-30T09:39:51Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:SISO]]
[[Category:Differential_algebraic_system]]

{{Infobox
|Title = RCL Circuit Equations
|Benchmark ID =
* rclCircuitEquations_n1841m16q16
* rclCircuitEquations_n306m2q2
|Category = oberwolfach
|System-Class = LTI-FOS
|nstates =
* 1841
* 306
|ninputs =
* 16
* 2
|noutputs =
* 16
* 2
|nparameters = 0
|components = A, B, C, E
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
|Zenodo-link = NA
}}

==Description==
These benchmark originate from [[wikipedia:Very_Large_Scale_Integration|VLSI]] circuits.
Specifically [[wikipedia:RLC_circuit|resistor-capacitor-inductor]] circuits, which can be represented by first order descriptor systems,
following a modeling process based on the two [[wikipedia:Kirchhoff's_circuit_laws|Kirchhoff's circuit laws]] and the [[wikipedia:Modified_nodal_analysis| branch constitutive relations]].

===PEEC Problem===
This RCL circuit is a [[wikipedia:Partial_element_equivalent_circuit|PEEC]] discretization<ref name="ruehli1974"/> and has 2100 capacitors, 172 inductors, 6990 inductive couplings, as well as a resistive source<ref name="feldmann1995"/>,<ref name="freund2005"/>.
The resulting model has 306 states, and two inputs and outputs.

===Package Problem===
The second problem models a 64-pin package of an [[wikipedia:RF_circuit|RF]] circuit.
A subset of eight pins carry signals, which leads to sixteen terminals (eight interior and eight exterior)<ref name="bai1997"/>,<ref name="freund2005"/>.
The resulting model has 1841 states, and sixteen inputs and outputs.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; see <ref name="freund2005"/>.

==Data==

The PEEC problem and package problem are available as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] files, providing the <math>A</math>, <math>B</math>, <math>E</math> matrices,
while <math>C = B^\intercal</math> is assumed:

* [[Media:PEEC.zip|PEEC.zip]] (32.8KB)
* [[Media:Package.zip|Package.zip]] (78.7KB)

==Dimensions==

System structure:

:<math>
\begin{align}
E \dot{x}(t) &= Ax(t) + Bu(t) \\
y(t) &= Cx(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}^{M \times N}</math>.

System variants:

<math>N = 306</math>, <math>M = 2</math>, for the PEEC problem, and <math>N = 1841</math>, <math>M = 16</math> for the package problem.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''RCL Circuit Equations'''. MORwiki - Model Order Reduction Wiki, 2019. http://modelreduction.org/index.php/RCL_Circuit_Equations

@MISC{morwiki_convection,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {RCL Circuit Equations},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/RCL_Circuit_Equations}</nowiki>,
year = {20XX}
}

* For the background on the benchmark:

@INCOLLECTION{morFre05,
author = <nowiki>{R.W. Freund}</nowiki>,
title = {RCL Circuit Equations},
booktitle = {Dimension Reduction of Large-Scale Systems},
pages = {367--371),
year = {2005},
doi = {10.1007/3-540-27909-1_22}
}

==References==

<references>

<ref name="ruehli1974">A.E. Ruehli, [https://doi.org/10.1109/TMTT.1974.1128204 Equivalent Circuit Models for Three-Dimensional Multiconductor Systems], IEEE Transactions on Microwave Theory and Techniques 22(1): 216--221, 1974.</ref>

<ref name="feldmann1995"> P. Feldmann, R.W. Freund , [https://doi.org/10.1109/43.384428 Efficient linear circuit analysis by Pade approximation via the Lanczos process], IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 14(5): 639--649, 1995.</ref>

<ref name="bai1997">Z. Bai, P. Feldmann, R.W. Freund, [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.34.7763&rep=rep1&type=pdf Equivalent Stable and Passive Reduced-Order Models Based on Partial Pade Approximation Via the Lanczos Process], Numerical Analysis Manuscript 97(3): 1--17, 1997.</ref>

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

<ref name="freund2005"> R.W. Freund, [https://doi.org/10.1007/3-540-27909-1_22 RCL Circuit Equations], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 367--371, 2005.</ref>

</references>

Random

2023-11-30T09:39:29Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:Sparse]]
[[Category:SISO]]

{{Infobox
|Title = Random
|Benchmark ID = random_n200m1q1
|Category = slicot
|System-Class = LTI-FOS
|nstates = 200
|ninputs = 1
|noutputs = 1
|nparameters = 0
|components = A, B, C
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
|Zenodo-link = NA
}}

==Description: Random Example==

This benchmark models is a randomly generated system.
More details can be found in <ref name="declippel97"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

==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> 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/random.zip random.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

==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}^{200 \times 200}</math>,
<math>B \in \mathbb{R}^{200 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 200}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MANUAL{slicot_rand,
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 background on the benchmark:

@MASTERSTHESIS{DeC97,
author = <nowiki>{V. De Clippel}</nowiki>,
title = {Mod\`eles R\'eduits de Grands Syst\`emes Dynamiques},
school = {Universit\'e Catholique de Louvain},
year = {1997}
}

==References==

<references>

<ref name="declippel97"> V. De Clippel. Modeles Reduits de Grands Systemes Dynamiques. Universite catholique de Louvain, 1997.</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

2023-11-30T09:39:02Z

Malhotra: infobox string

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

{{Infobox
|Title = Plate with tuned vibration absorbers
|Benchmark ID =
* plateTVA_n201900m1q1
* plateTVA_n201900m1q28278
|Category = misc
|System-Class = LTI-SOS
|nstates = 201900
|ninputs = 1
|noutputs =
* 1
* 28278
|nparameters = 0
|components = B, C, E, K, M
|License = Creative Commons Attribution 4.0 International
|Creator = [[User:Aumann]]
|Editor =
* [[User:Aumann]]
|Zenodo-link =
* https://zenodo.org/record/7671686/files/plateTVA_n201900m1q1.mat
* https://zenodo.org/record/7671686/files/plateTVA_n201900m1q28278.mat
}}

<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>

Penzl's FOM

2023-11-30T09:38:40Z

Malhotra: infobox string

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

{{Infobox
|Title = Penzl's FOM
|Benchmark ID = penzlFOM_n1006m1q1
|Category = slicot
|System-Class = LTI-FOS
|nstates = 1006
|ninputs = 1
|noutputs = 1
|nparameters = 0
|components = A, B, C
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
* [[User:Yue]]
|Zenodo-link = NA
}}

==Description==

This benchmark is an artificial example system of order <math>1006</math> from <ref name="penzl06"/> also listed in <ref name="chahlaoui02"/>. It has long been regarded as a standard "full order model" (FOM) for testing new methods.

The benchmark system consists of the following system components:

<math>
\begin{array}{rcl}
A &=& \begin{pmatrix} A_1 \\ & A_2 \\ & & A_3 \\ & & & A_4 \end{pmatrix}, \\
A_1 &=& \begin{pmatrix} -1 & 100 \\ -100 & -1 \end{pmatrix}, \;
A_2 = \begin{pmatrix} -1 & 200 \\ -200 & -1 \end{pmatrix}, \;
A_3 = \begin{pmatrix} -1 & 400 \\ -400 & -1 \end{pmatrix}, \;
A_4 = \begin{pmatrix} -1 \\ & -2 \\ & & \ddots \\ & & & -1000 \end{pmatrix}, \\
B &=& \begin{pmatrix} 10 & 10 & 10 & 10 & 10 & 10 & 1 & \dots & 1 \end{pmatrix}^T, \\
C &=& B^T.
\end{array}
</math>

This system is a theoretical construct, but features a non-smooth [[wikipedia:Bode_plot|Bode plot]] with three spikes.

===MIMO Variant===

In <ref name="heyouni08"/> a MIMO variant of this benchmark is utilized by adding random vectors to <math>B</math> and <math>C</math>.

===Parametric Variant===

In <ref name="Ionita14"/>, a parametric variant of this benchmark is formulated by redefining
<math>
A_1 = \begin{pmatrix} -1 & p \\ -p & -1 \end{pmatrix}.
</math>

==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> 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/fom.zip fom.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

==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}^{1006 \times 1006}</math>,
<math>B \in \mathbb{R}^{1006 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 1006}</math>,

==Citation==

To cite this benchmark, use the following references:

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

@MANUAL{slicot_fom,
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 background on the benchmark:

@ARTICLE{morPen06,
author = <nowiki>{T. Penzl}</nowiki>,
title = {Algorithms for Model Reduction of Large Dynamical Systems},
journal = {Linear Algebra and its Application},
volume = {415},
number = {2--3},
pages = {322--343},
year = {2006},
doi = {10.1016/j.laa.2006.01.007}
}

==References==

<references>

<ref name="penzl06"> T. Penzl. [https://doi.org/10.1016/j.laa.2006.01.007 Algorithms for Model Reduction of Large Dynamical Systems]. Linear Algebra and its Application 415(2--3): 322--343, 2006.</ref>

<ref name="heyouni08"> M. Heyouni, K. Jbilou, A. Messaoudi, K. Tabaa. [https://www.scielo.br/j/cam/a/Sq6GFZqcXSwNQKk3SmpB46p/?lang=en Model Reduction in Large-Scale MIMO Dynamical Systems via the Block Lanczos Method]. Computational & Applied Mathematics 27(11): 211--236, 2008.</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>

<ref name="Ionita14"> A. C. Ionita,A. C. Antoulas, [https://doi.org/10.1137/130914619 Data-Driven Parametrized Model Reduction in the Loewner Framework], SIAM J. Sci. Comput. 36(3): A984–A1007, 2014.</ref>

</references>

Peek Inductor

2023-11-30T09:38:21Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Oberwolfach]]

{{Infobox
|Title = Peek Inductor
|Benchmark ID = peekInductor_n1434m1q1
|Category = oberwolfach
|System-Class = LTI-FOS
|nstates = 1434
|ninputs = 1
|noutputs = 1
|nparameters = 0
|components = A, B, C, E
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
|Zenodo-link = NA
}}

==Description: Spiral Inductor PEEC Model==
<figure id="fig1">[[File:Peek1.jpg|490px|thumb|right|<caption>Spiral inductor with part of overhanging copper plane</caption>]]</figure>

The description of the [[wikipedia:Partial_element_equivalent_circuit|PEEC]] model of a spiral inductor can be found in [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/PeekInductor.pdf PeekInductor.pdf].

The complex impedance is:

:<math>
Z(w) = Resis(w)+i*w*Induc(w) = G(i*w)^{-1}=(B^\intercal(-A+i*w*E)^{-1}B)^{-1}
</math>

A plots of <math>Resis(w)</math> can be found in [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/PeekInductor-Rspiral_skin.pdf PeekInductor-Rspiral_skin.pdf] and a plot of <math>Induc(w)</math> in [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/PeekInductor-Lspiral_skin.pdf PeekInductor-Lspiral_skin.pdf].

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38891, see <ref name="li2005"/>.

==Data==

The model is of order <math>N=1434</math> and of the form:
:<math>
\begin{array}{rcl}
E \dot{x}(t) &=& Ax(t) + Bu(t) \\
y(t) &=& B^\intercal x(t)
\end{array}
</math>

and can be downloaded as [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/PeekInductor-dim1e3-spiral_inductor_peec.tar.gz PeekInductor-dim1e3-spiral_inductor_peec.tar.gz] (10.5 MB).

Short [[wikipedia:MATLAB|Matlab]] files to:

* plot <math>Resis(w)</math> and <math>Induc(w)</math>,
* perform a [[PRIMA]] reduction of order 50,
* produce symmetrized standard state-space system: <math>\dot{x}(t) = A_{symm}x(t)+ B_{symm}u(t)</math>, <math>y(t) = B_{symm}^\intercal x(t)</math>, where <math>A_{symm}</math> is symmetric.

can be found in [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/PeekInductor-plot_spiral.tar.gz PeekInductor-plot_spiral.tar.gz]

==Dimensions==

System structure:

:<math>
\begin{align}
E \dot{x}(t) &= Ax(t) + Bu(t) \\
y(t) &= B^\intercal x(t)
\end{align}
</math>

System dimensions:

<math>E \in \mathbb{R}^{1434 \times 1434}</math>,
<math>A \in \mathbb{R}^{1434 \times 1434}</math>,
<math>B \in \mathbb{R}^{1434 \times 1}</math>.

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

@INCOLLECTION{morLiK05,
author = <nowiki>{J.R. Li, M. Kamon}</nowiki>,
title = {PEEC Model of a Spiral Inductor Generated by Fasthenry},
booktitle = {Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering},
volume = {45},
pages = {373--377},
year = {2005},
doi = {10.1007/3-540-27909-1_23}
}

==References==

<references>

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

<ref name="li2005">J.R. Li, M. Kamon, [https://doi.org/10.1007/3-540-27909-1_23 PEEC Model of a Spiral Inductor Generated by Fasthenry]. In: Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol 45: 373--377, 2005.</ref>

</references>

PEEC Model (SLICOT)

2023-11-30T09:37:59Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:SISO]]
[[Category:Sparse]]

{{Infobox
|Title = PEEC Model (SLICOT)
|Benchmark ID = peecModel_n480m1q1
|Category = slicot
|System-Class = LTI-FOS
|nstates = 480
|ninputs = 1
|noutputs = 1
|nparameters = 0
|components = A, B, C, E
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
|Zenodo-link = NA
}}

==Description: Partial Element Equivalent Circuit Model==

This benchmark models a [[wikipedia:Partial_element_equivalent_circuit|PEEC]] of patch antenna structure.
More details can be found in <ref name="heeb92"/>, <ref name="chiprout94"/>, <ref name="grimme97"/> and <ref name="chahlaoui02"/>.

For a larger PEEC model see the alternative [[Peek_Inductor|Peek Inductor]] benchmark.

==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>, <math>E</math> 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/peec.zip peec.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

==Dimensions==

System structure:

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

System dimensions:

<math>A \in \mathbb{R}^{480 \times 480}</math>,
<math>B \in \mathbb{R}^{480 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 480}</math>,
<math>E \in \mathbb{R}^{480 \times 480}</math>

==Citation==

To cite this benchmark, use the following references:

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

@MANUAL{slicot_peec,
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 background on the benchmark:

@INPROCEEDINGS{HeeRBetal92,
author = <nowiki>{H. Heeb and A.E. Ruehli and J.E. Bracken and R.A. Rohrer}</nowiki>,
title = {Three Dimensional Circuit Oriented Electromagnetic Modeling for VLSI Interconnects},
booktitle = {Proceedings of the 1992 IEEE International Conference on Computer Design: VLSI in Computers \& Processors},
pages = {218--221},
year = {1992},
doi = {10.1109/ICCD.1992.276253}
}

==References==

<references>

<ref name="chiprout94"> E. Chiprout, M.S. Nakhla. [https://doi.org/10.1007/978-1-4615-3116-6 Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis]. The Springer International Series in Engineering and Computer Science, vol 252, Springer, 1999.</ref>

<ref name="grimme97"> E.J. Grimme. [http://hdl.handle.net/2142/81180 Krylov Projection Methods for Model Reduction]. PhD Thesis, University of Illinois at Urbana-Champaign, 1998.</ref>

<ref name="heeb92"> H. Heeb, A.E. Ruehli, J.E. Bracken, R.A. Rohrer. [https://doi.org/10.1109/ICCD.1992.276253 Three Dimensional Circuit Oriented Electromagnetic Modeling for VLSI Interconnects]. Proceedings of the 1992 IEEE International Conference on Computer Design: VLSI in Computers & Processors: 218--221, 1992.</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>

Orr-Sommerfeld

2023-11-30T09:37:39Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:SISO]]
[[Category:Dense]]

{{Infobox
|Title = Orr-Sommerfeld
|Benchmark ID = orrSommerfeld_n100m1q1
|Category = slicot
|System-Class = LTI-FOS
|nstates = 100
|ninputs = 1
|noutputs = 1
|nparameters = 0
|components = A, B, C
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
|Zenodo-link = NA
}}

==Description: Orr-Sommerfeld Operator for Couette Flow==

This benchmark models Orr-Sommerfeld Operator for a [[wikipedia:Couette_flow|Couette flow]].
More details can be found in <ref name="farrell01"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

==Origin==

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

==Data==

The system matrix <math>A</math> is 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/Orr-Som.zip Orr-Som.zip] and is stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

Note that the data file only contains an <math>A</math> matrix! For <math>B = C^\intercal
</math> any normalized vector <math>B^\intercal B = 1</math>, can be chosen.

==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}^{100 \times 100}</math>,
<math>B \in \mathbb{R}^{100 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 100}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MANUAL{slicot_orrs,
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 background on the benchmark:

@ARTICLE{FarI01,
author = <nowiki>{B.F. Farrell and P.J. Ioannou}</nowiki>,
title = {Accurate Low-Dimensional Approximation of the Linear Dynamics of Fluid Flow},
journal = {Journal of the Atmospheric Sciences},
volume = {58},
number = {18},
pages = {2771--2789},
year = {2001},
doi = {10.1175/1520-0469(2001)058<2771:ALDAOT>2.0.CO;2}
}

==References==

<references>

<ref name="farrell01"> B.F. Farrell, P.J. Ioannou. [https://doi.org/10.1175/1520-0469(2001)058%3C2771:ALDAOT%3E2.0.CO;2 Accurate Low-Dimensional Approximation of the Linear Dynamics of Fluid Flow]. Journal of the Atmospheric Sciences, 58(18): 2771--2789, 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>

Modified Gyroscope

2023-11-30T09:36:43Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:ODE]]
[[Category:linear]]
[[Category:Parametric]]
[[Category:affine parameter representation]]
[[Category:time invariant]]
[[Category:second differential order]]

{{Infobox
|Title = Modified Gyroscope
|Benchmark ID = modifiedGyroscope_n17931m1q1
|Category = misc
|System-Class = AP-LTI-SOS
|nstates = 17931
|ninputs = 1
|noutputs = 1
|nparameters = 3
|components = B, C, E, K, M
|License = NA
|Creator = [[User:Feng]]
|Editor =
* [[User:Feng]]
* [[User:Himpe]]
* [[User:Will]]
* [[User:Saak]]
* [[User:Lund]]
|Zenodo-link = NA
}}

==Description==
<figure id="fig:gyro">[[File:Gyroscope.jpg|300px|thumb|right|<caption>Schematic representation of the gyroscope.</caption>]]</figure>

The device is a [[wikipedia:Vibrating_structure_gyroscope#MEMS_gyroscopes|MEMS gyroscope]] based on the butterfly gyroscope <ref name="lienemann2004"/> developed at the [http://www.imego.com/ Imego institute] in Gothenburg,
Sweden (see also: [[Butterfly_Gyroscope|Butterfly Gyroscope]], where a non-parametrized model for the device is given).
A [[wikipedia:Gyroscope|gyroscope]] is a device used to measure angular rates in up to three axes.

The basic working principle of the '''gyroscope''' can be described as follows, see also <ref name="Moo07"/>.
Without applied external rotation, the paddles vibrate in phase with the function <math>z(t),</math> see Fig. 1.
Under the influence of an external rotation about the <math>x</math>-axis (drawn in red),
an additional force due to the Coriolis acceleration acts upon the paddles.
This force leads to an additional small out-of-phase vibration between two paddles on the same side of the bearing.
This out-of phase vibration is measured as the difference of the <math>z</math>-displacement of the nodes with the red dots.
Thus, measuring the displacement of two adjacent paddles, the rotation velocity can be ascertained.

==Motivation==

When planning for and making decisions on future improvements of the butterfly gyroscope, it is of importance to improve the efficiency of the gyro simulations. Repeated analysis of the sensor structure have to be conducted with respect to a number of important issues.
Examples of such are sensitivity to shock, linear and angular vibration sensitivity, reaction to large rates and/or acceleration,
different types of excitation load cases and the effect of force-feedback.
The use of model order reduction indeed decreases run time for repeated simulations.

==The Parametrized Model==

Two parameters are of special interest for the model.
The first one is the quantity that is to be sensed, the rotation velocity <math>\theta</math> around the <math>x</math>-axes.
The second parameter is the width of the bearing, <math>d</math>.
The parametrized system below is obtained by finite element discretization of the parametrized model (in the form of partial differential equations) for the '''gyroscope'''.
The details of constructing the parametrized system can be found in <ref name="Moo07"/>.
The system is of the following form:

:<math>
\begin{align}
M(d)\ddot{x}(t) +D(d,\theta)\dot{x}(t) +T(d)x(t) &= B, \\
y(t) &=Cx(t).
\end{align}
</math>

Here,

* <math>M(d)=M_1 + dM_2\in \mathbb R^{n\times n}</math> is the mass matrix,
* <math>D(d,\theta)=\theta(D_1 + d D_2)\in \mathbb R^{n\times n}</math> is the damping matrix,
* <math>T(d)=T_1+(1/d)T_2+dT_3\in \mathbb R^{n\times n}</math> is the stiffness matrix,
* <math>B \in \mathbb R^{n \times 1}</math> is the load vector,
* <math>C \in \mathbb R^{1 \times n}</math> is the output matrix,
* <math>x \in \mathbb R^{n}</math> is the state vector,
* and <math>y \in \mathbb R </math> is the output response.

The quantity of interest <math>y</math> of the system is <math>\delta z(t)</math>, which is the difference of the displacement <math>z(t)</math> between the two red markings on the ''east'' side of the bearing (see <xr id="fig:gyro"/>).

The parameters of the system, <math>d</math> and <math>\theta</math>, represent the width of the bearing(<math>d</math>) and the rotation velocity along the <math>x</math>-axis (<math>\theta</math>), with the ranges: <math>\theta\in [10^{-7}, 10^{-5}]</math> and <math>d\in [1,2]</math>.

The device works in the frequency range <math>f \in [0.025, 0.25]</math>MHz and the degrees of freedom are <math>n = 17913</math>.

==Data==

The model is generated in ANSYS.
The system matrices <math>M_1, \, M_2, \, D_1, \, D_2, \, T_1, \, T_2, \, T_3</math>, and <math> B</math> are in the [http://math.nist.gov/MatrixMarket/ MatrixMarket format], and can be downloaded here: [[Media: Gyroscope_modi.tgz|Gyroscope_modi.tgz]]. The matrix <math>C</math> defines the output, which has zeros on all the entries, except on the 2315th entry, where the value is <math>-1</math>, and on the 5806th entry, the value is <math>1</math>;
in MATLAB notation, it is <tt>C(1,2315) = -1</tt> and <tt>C(1,5806) = 1</tt>.

The matrices have been reindexed for [[MORB]], as shown in the next section. In particular, <math>D_i</math> is denoted instead by <math>E_i</math>, and <math>T_i</math> by <math>K_i</math>.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
(M_1 + d M_2)\ddot{x}(t) + \theta(E_1 + d E_2) \dot{x}(t) + (K_1 + d^{-1} K_2 + d K_3)x(t) &=& B \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>M_{1,2}, E_{1,2}, K_{1,2,3} \in \mathbb{R}^{17931 \times 17931}</math>,
<math>B \in \mathbb{R}^{17931 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 17931}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_modgyro,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Modified Gyroscope},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Modified_Gyroscope}</nowiki>,
year = 2018
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morMoo07 morMoo07] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morMoo07 BibTeX])

==References==

<references>

<ref name="lienemann2004">J. Lienemann, D. Billger, E. B. Rudnyi, A. Greiner, J. G. Korvink, "[http://www.nsti.org/procs/Nanotech2004v2/6/W58.01 MEMS Compact Modeling Meets Model Order Reduction: Examples of the Application of Arnoldi Methods to Microsystem Devices]", TechConnect Briefs (Technical Proceedings of the 2004 NSTI Nanotechnology Conference and Trade Show, Volume 2): 303--306, 2004.</ref>

<ref name="Moo07">C. Moosmann, "[http://www.freidok.uni-freiburg.de/volltexte/3971/ ParaMOR Model Order Reduction for parameterized MEMS applications]", PhD thesis, Department of Microsystems Engineering, University of Freiburg, Freiburg, Germany 2007.</ref>

</references>

==Contact==

'' [[User:Feng]] ''

Modified Nodal Analysis

2023-11-30T09:36:14Z

Malhotra: infobox string

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

{{Infobox
|Title = Modified Nodal Analysis
|Benchmark ID =
* mna_n10913m9q9
* mna_n4863m22q22
* mna_n578m9q9
* mna_n9223m18q18
* mna_n980m4q4
|Category = slicot
|System-Class = LTI-FOS
|nstates =
* 10913
* 4863
* 578
* 9223
* 980
|ninputs =
* 9
* 22
* 9
* 18
* 4
|noutputs =
* 9
* 22
* 9
* 18
* 4
|nparameters = 0
|components = A, B, C, E
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
|Zenodo-link = NA
}}

{{Infobox
|Title = Modified Gyroscope
|Benchmark ID = modifiedGyroscope_n17931m1q1
|Category = misc
|System-Class = AP-LTI-SOS
|nstates = 17931
|ninputs = 1
|noutputs = 1
|nparameters = 3
|components = B, C, E, K, M
|License = NA
|Creator = [[User:Feng]]
|Editor =
* [[User:Feng]]
* [[User:Himpe]]
* [[User:Will]]
* [[User:Saak]]
* [[User:Lund]]
|Zenodo-link = NA
}}

==Description: Modified Nodal Analysis Model==

This set of benchmark uses models resulting from [[wikipedia:Modified_nodal_analysis|modified nodal analysis]].
More details can be found in <ref name="odabasioglu98"/> and <ref name="chahlaoui02"/>.

==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>, <math>E</math> are available from the [http://slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page:

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Benchmark files and dimensions.''
|-
!
!Inputs
!States
!Outputs
|-
|[http://slicot.org/objects/software/shared/bench-data/MNA_1.zip MNA_1.zip]
|<math>M = 9</math>
|<math>N = 578</math>
|<math>Q = 9</math>
|-
|[http://slicot.org/objects/software/shared/bench-data/MNA_2.zip MNA_2.zip]
|<math>M = 18</math>
|<math>N = 9223</math>
|<math>Q = 18</math>
|-
|[http://slicot.org/objects/software/shared/bench-data/MNA_3.zip MNA_3.zip]
|<math>M = 22</math>
|<math>N = 4863</math>
|<math>Q = 22</math>
|-
|[http://slicot.org/objects/software/shared/bench-data/MNA_4.zip MNA_4.zip]
|<math>M = 4</math>
|<math>N = 980</math>
|<math>Q = 4</math>
|-
|[http://slicot.org/objects/software/shared/bench-data/MNA_5.zip MNA_5.zip]
|<math>M = 9</math>
|<math>N = 10913</math>
|<math>Q = 9</math>
|}

and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
E\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 M}</math>,
<math>C \in \mathbb{R}^{Q \times N}</math>,
<math>E \in \mathbb{R}^{N \times N}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MANUAL{slicot_mna,
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 background on the benchmark:

@ARTICLE{morOda98,
author = <nowiki>{A. Odabasioglu and M. Celik and L.T. Pileggi}</nowiki>,
title = {{PRIMA}: Passive Reduced-order Interconnect Macromodeling Algorithm},
journal = {IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems},
volume = {17},
number = {8},
pages = {645--654},
year = {1998},
doi = {10.1109/43.712097}
}

==References==

<references>

<ref name="odabasioglu98">A. Odabasioglu, M. Celik, L.T. Pileggi, [https://doi.org/10.1109/43.712097 PRIMA: Passive Reduced-order Interconnect Macromodeling Algorithm]. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 17(8): 645--654, 1998.</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>

Modified Nodal Analysis

2023-11-30T09:35:26Z

Malhotra: infobox string

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

{{Infobox
|Title = Modified Gyroscope
|Benchmark ID = modifiedGyroscope_n17931m1q1
|Category = misc
|System-Class = AP-LTI-SOS
|nstates = 17931
|ninputs = 1
|noutputs = 1
|nparameters = 3
|components = B, C, E, K, M
|License = NA
|Creator = [[User:Feng]]
|Editor =
* [[User:Feng]]
* [[User:Himpe]]
* [[User:Will]]
* [[User:Saak]]
* [[User:Lund]]
|Zenodo-link = NA
}}

==Description: Modified Nodal Analysis Model==

This set of benchmark uses models resulting from [[wikipedia:Modified_nodal_analysis|modified nodal analysis]].
More details can be found in <ref name="odabasioglu98"/> and <ref name="chahlaoui02"/>.

==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>, <math>E</math> are available from the [http://slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page:

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Benchmark files and dimensions.''
|-
!
!Inputs
!States
!Outputs
|-
|[http://slicot.org/objects/software/shared/bench-data/MNA_1.zip MNA_1.zip]
|<math>M = 9</math>
|<math>N = 578</math>
|<math>Q = 9</math>
|-
|[http://slicot.org/objects/software/shared/bench-data/MNA_2.zip MNA_2.zip]
|<math>M = 18</math>
|<math>N = 9223</math>
|<math>Q = 18</math>
|-
|[http://slicot.org/objects/software/shared/bench-data/MNA_3.zip MNA_3.zip]
|<math>M = 22</math>
|<math>N = 4863</math>
|<math>Q = 22</math>
|-
|[http://slicot.org/objects/software/shared/bench-data/MNA_4.zip MNA_4.zip]
|<math>M = 4</math>
|<math>N = 980</math>
|<math>Q = 4</math>
|-
|[http://slicot.org/objects/software/shared/bench-data/MNA_5.zip MNA_5.zip]
|<math>M = 9</math>
|<math>N = 10913</math>
|<math>Q = 9</math>
|}

and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
E\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 M}</math>,
<math>C \in \mathbb{R}^{Q \times N}</math>,
<math>E \in \mathbb{R}^{N \times N}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MANUAL{slicot_mna,
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 background on the benchmark:

@ARTICLE{morOda98,
author = <nowiki>{A. Odabasioglu and M. Celik and L.T. Pileggi}</nowiki>,
title = {{PRIMA}: Passive Reduced-order Interconnect Macromodeling Algorithm},
journal = {IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems},
volume = {17},
number = {8},
pages = {645--654},
year = {1998},
doi = {10.1109/43.712097}
}

==References==

<references>

<ref name="odabasioglu98">A. Odabasioglu, M. Celik, L.T. Pileggi, [https://doi.org/10.1109/43.712097 PRIMA: Passive Reduced-order Interconnect Macromodeling Algorithm]. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 17(8): 645--654, 1998.</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>

Micropyros Thruster

2023-11-30T09:34:52Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:linear]]
[[Category:First differential order]]
[[Category:Sparse]]
[[Category:Time invariant]]
[[Category:MIMO]]
[[Category:ODE]]

{{Infobox
|Title = Micropyros Thruster
|Benchmark ID =
* micropyrosThruster_n11445m1q7
* micropyrosThruster_n20360m1q7
* micropyrosThruster_n4257m1q7
* micropyrosThruster_n79171m1q7
|Category = oberwolfach
|System-Class = LTI-FOS
|nstates =
* 11445
* 20360
* 4257
* 79171
|ninputs = 1
|noutputs = 7
|nparameters = 0
|components = A, B, C, E
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
* [[User:Stillfjord]]
* [[User:Saak]]
|Zenodo-link = NA
}}

==Description==
<figure id="fig1">[[File:Micropyros.jpg|490px|thumb|right|Figure 1]]</figure>

The goal of the European project [https://web.archive.org/web/20080321055834/http://www.laas.fr:80/Micropyros/ Micropyros] was to develop a microthruster array shown in Fig. 1.
It is based on the co-integration of solid fuel with a silicon micromachined system.
In addition to space applications, the device can be also used for gas generation or as a highly-energetic actuator.
When the production of a bit-impulse is required, the fuel is ignited by heating a resistor at the top of a particular microthruster.
Design requirements and modeling alternatives are described in <ref name="rudnyi02"/>.
The discussion of electro-thermal modeling related to the benchmark can be found in <ref name="korvink03"/>.

The benchmark contains a simplified thermal model of a single microthruster to help with a design problem to reach the ignition temperature within the fuel and at the same time not to reach the critical temperature at neighboring microthrusters, that is, at the border of the computational domain.
At the same time, the resistor temperature during the heating pulse should not become too high as this leads to the destruction of the membrane.

The benchmark suite has been made with the Micropyros software developed by [http://www.imtek.uni-freiburg.de/professuren/simulation/simulation IMTEK].
There are four different test cases described in Table 1 with the goal to cover different cases of different computational complexity.
Note that the results from different models cannot be compared directly with each other as the output nodes are located in slightly different geometrical positions and there is some difference in modeling for the 3D and 2D-axisymmetric cases.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Microthruster benchmarks.''
|-
|Code
|Comment
|Dimension
|nnz(A)
|nnz(E)
|-
|T2DAL
|2D-axisymmetric, linear elements
|style="text-align:right;" |<math>4\,257</math>
|style="text-align:right;" |<math>20\,861</math>
|style="text-align:right;" |<math>4\,257</math>
|-
|T2DAH
|2D-axisymmetric, quadratic elements
|style="text-align:right;" |<math>11\,445</math>
|style="text-align:right;" |<math>93\,781</math>
|style="text-align:right;" |<math>93\,781</math>
|-
|T3DL
|3D, linear elements
|style="text-align:right;" |<math>20\,360</math>
|style="text-align:right;" |<math>265\,113</math>
|style="text-align:right;" |<math>20\,360</math>
|-
|T3DH
|3D, quadratic elements
|style="text-align:right;" |<math>79\,171</math>
|style="text-align:right;" |<math>2\,215\,638</math>
|style="text-align:right;" |<math>2\,215\,638</math>
|}

The device solid model has been made and meshed in [http://www.ansys.com/ ANSYS].
The material properties assumed to be constant. Temperature is assumed to be in Celsius with the initial state of <math>0</math>Celsius.

The output nodes are described in Table 2. Nodes 2 to 5 show the fuel temperature distribution and nodes 6 and 7 characterize temperature in the wafer, nodes 5 and 7 being the most far away from the resistor.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 2: Outputs for the microthruster models.''
|-
|#
|Code
|Comment
|-
|1
|aHeater
|within the heater
|-
|2
|FuelTop
|fuel just below the heater
|-
|3
|FT-100
|fuel 0.1 mm below the heater
|-
|4
|FT-200
|fuel 0.2 mm below the heater
|-
|5
|FuelBot
|fuel bottom
|-
|6
|WafTop1
|wafer top (touching fuel)
|-
|7
|WafTop2
|wafer top (end of computational domain)
|}

The benchmark files contain a constant load vector, corresponding to the constant power input of <math>150</math>mW.
In order to insert a weak nonlinearity related to the dependence of the resistivity on temperature, one has to multiply the load vector by a function

<math>
f(T) = 1 + 9 \cdot 10^{-4} \cdot T_1 + 3 \cdot 10^{-7} \cdot T_1^2
</math>

assuming the constant current. The Temperature <math>T_1</math> in the equation above, is the temperature at the node 1.

The first order ordinary differential equations are written as

<math>
\begin{array}{rcl}
E \frac{\partial}{\partial t} T(t) &=& A T(t) + f(T) B u(t)\\
y(t) &=& C T(t)
\end{array}
</math>

where <math>E</math> and <math>A</math> are the system matrices (both are symmetric), <math>B</math> is the load vector, <math>C</math> is the output matrix, and <math>T</math> is the vector of unknown temperatures.
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 1}</math>,
<math>C \in \mathbb{R}^{7 \times N}</math>

The ANSYS results for the original models as well as the reduced models obtained by [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem] can be found at the micropyros page:
choose EleThermo for T2DAL and T2DAH or EleThermo3D for T3DL and T3DH.
The system matrices have been converted to the [http://math.nist.gov/MatrixMarket/ Matrix Market] format by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem].

The model reduction of the microthruster model by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem] is described in <ref name="bechthold03"/>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>, No. 38847.

==Data==

Download matrices in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format:
* [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/MicropyrosThruster-dim1e3-T2DAL.tar.gz MicropyrosThruster-dim1e3-T2DAL.tar.gz] (215.7 kB)
* [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/MicropyrosThruster-dim1e4-T2DAH.tar.gz MicropyrosThruster-dim1e4-T2DAH.tar.gz] (1.6 MB)
* [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/MicropyrosThruster-dim1e4-T3DL.tar.gz MicropyrosThruster-dim1e4-T3DL.tar.gz] (2.1 MB)
* [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/MicropyrosThruster-dim1e5-T3DH.tar.gz MicropyrosThruster-dim1e5-T3DH.tar.gz] (36.7 MB)

The matrix name is used as an extension of the matrix file. File <tt>*.C.names</tt> contains a list of output names written consecutively.

==Dimensions==

System structure:

:<math>
\begin{align}
E \dot{x}(t) &= Ax(t) + B u(t)\\
y(t) &= Cx(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 1}</math>,
<math>C \in \mathbb{R}^{7 \times N}</math>.

System variants:

<tt>T2DAL</tt>: <math>N = 4\,257</math>,
<tt>T2DAH</tt>: <math>N = 11\,445</math>,
<tt>T3DL</tt>: <math>N = 20\,360</math>,
<tt>T3DH</tt>: <math>N = 79\,171</math>.

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

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection, '''Micropyros Thruster'''. hosted at MORwiki - Model Order Reduction Wiki, 2005. http://modelreduction.org/index.php/Micropyros_Thruster

@MISC{morwiki_thruster,
author = <nowiki>{{Oberwolfach Benchmark Collection}}</nowiki>,
title = {Micropyros Thruster},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Micropyros_Thruster}</nowiki>,
year = 2005
}

* For the background on the benchmark:

@InProceedings{RudBKetal02,
author = {E.B. Rudnyi and T. Bechtold and J.G. Korvink and
C. Rossi},
title = {Solid Propellant Microthruster: Theory of Operation
and Modelling Strategy},
booktitle = {Nanotech 2002 - At the Edge of Revolution, September
9--12, 2002, Houston (USA)},
year = 2002,
note = {AIAA Paper 2002-5755},
doi = {10.2514/6.2002-5755}
}

==References==

<references>

<ref name="rudnyi02"> E.B. Rudnyi, T. Bechtold, J.G. Korvink, C. Rossi, [https://doi.org/10.2514/6.2002-5755 Solid Propellant Microthruster: Theory of Operation and Modelling Strategy], Nanotech 2002 - At the Edge of Revolution, September 9--12, 2002, Houston (USA) AIAA Paper 2002-5755.</ref>

<ref name="korvink03"> G. Korvink, E.B. Rudnyi, [http://modelreduction.com/doc/papers/korvink03EUROSIME.pdf Computer-aided engineering of electro-thermal MST devices: moving from device to system simulation], EUROSIME'03, 4th international conference on thermal & mechanical simulation and experiments in micro-electronics and micro-systems Aix-en-Provence (France), March 30 -- April 2, 2003.</ref>

<ref name="bechthold03"> T. Bechtold, E. B. Rudnyi, J. G. Korvink and C. Rossi, [http://modelreduction.com/doc/papers/rudnyi04MTNS.pdf Efficient Modelling and Simulation of 3D Electro-Thermal Model for a Pyrotechnical Microthruster], International Workshop on Micro and Nanotechnology for Power Generation and Energy Conversion Applications PowerMEMS 2003, Makuhari (Japan), December 4--5, 2003.</ref>

<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>

</references>

Nonlinear Heat Transfer

2023-11-30T09:34:19Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Oberwolfach]]

{{Infobox
|Title = Nonlinear Heat Transfer
|Benchmark ID =
* linearHeatTransfer_n15m2q2
* nonlinearHeatTransfer_n15m2q2
* nonlinearHeatTransfer_n410m2q2
|Category = oberwolfach
|System-Class =
* LTI-FOS
* NLTI-FOS
* NLTI-FOS
|nstates =
* 15
* 15
* 410
|ninputs = 2
|noutputs = 2
|nparameters = 0
|components =
* A, B, C, E
* A, B, C, E, F, f
* A, B, C, E, F, f
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
|Zenodo-link = NA
}}

==Description==

The simulation of heat transport for a single device is easily tackled by current computational resources, even for a complex, finely structured geometry;
however, the calculation of a multi-scale system consisting of a large number of those devices, e.g., assembled printed circuit boards, is still a challenge.
A further problem is the large change in heat conductivity of many semiconductor materials with temperature.
We model the heat transfer along a 1D beam that has a nonlinear heat capacity which is represented by a polynomial of arbitrary degree as a function of the temperature state.
For accurate modelling of the temperature distribution, the resulting model requires many state variables to be described adequately.
The resulting complexity, i.e., number of first order differential equations and nonlinear parts, is such that a simplification or model reduction is needed in order to perform a simulation in an acceptable amount of time for the applications at hand.
Thus the need for model order reduction emerges.

===Model description===

We model the heat transfer along a 1D beam with length <math>L</math>, cross sectional area <math>A</math> and nonlinear heat conductivity represented by a polynomial in temperature <math>T(x,t)</math> of arbitrary degree <math>n</math>:

<math>
\kappa(T) = a_0 + a_1 T + a_2 T^2 + \dots
</math>

The output of the model is the temperature <math>T(x,t)</math>, the degrees of freedom are the temperature from left to right.
The right end of the beam (at <math>x=L</math>) is fixed at ambient temperature <math>0</math>;
this node does not occur in the model any more.
The model features two inputs: The first one is a time-dependent uniform heat flux <math>f</math> [W/m2] flowing in from the left end (at <math>x=0</math>).
The second one is a time dependent heat source <math>Q</math> [W/m3] in the beam volume, e.g. from an electric current.

===Benchmark examples===

An interactive matrix generator has been created using [http://www.wolfram.com/ Wolfram Research]'s [http://www.wolfram.com/products/webmathematica webMathematica]. Models produced by this generator are in the <tt>DSIF</tt> format, which allows for nonlinear terms.

Three ready-made examples are available (all files are <tt>gzip</tt> compressed <tt>DSIF</tt> files, Units: SI):

====Linear example (heat conductivity not temperature dependent)====

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Specification of <tt>NonlinearHeatCond-n15-linear.zip</tt>.''
|-
|Property
|Symbol
|Unit
|Value
|-
|Number of nodes
|
|
|<math>15</math>
|-
|Beam length
|<math>l</math>
|[m]
|<math>0.1</math>
|-
|Cross-sectional area
|<math>A</math>
|[m2]
|<math>10^{-4}</math>
|-
|Material density
|<math>\rho</math>
|[kg/m3]
|<math>3970</math>
|-
|Heat capacity
|<math>C_p</math>
|[J/kg K]
|<math>766</math>
|-
|Heat conductivity
|<math>a_0</math>
|[W/m K]
|<math>36</math>
|}

====Nonlinear examples (heat conductivity temperature dependent)====

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Specification of <tt>NonlinearHeatCond-n15-nonlinear.zip</tt>.''
|-
|Property
|Symbol
|Unit
|Value
|-
|Number of nodes
|
|
|<math>15</math>
|-
|Beam length
|<math>l</math>
|[m]
|<math>0.1</math>
|-
|Cross-sectional area
|<math>A</math>
|[m2]
|<math>10^{-4}</math>
|-
|Material density
|<math>\rho</math>
|[kg/m3]
|<math>3970</math>
|-
|Heat capacity
|<math>C_p</math>
|[J/kg K]
|<math>766</math>
|-
|Heat conductivity
|<math>a_0</math>
|[W/m K]
|<math>36</math>
|-
|Heat conductivity
|<math>a1</math>
|[W/m K2]
|<math>-0.1116</math>
|-
|Heat conductivity
|<math>a_2</math>
|[W/m K3]
|<math>0.00017298</math>
|-
|Heat conductivity
|<math>a_3</math>
|[W/m K4]
|<math>-1.78746 \cdot 10^{-7}</math>
|-
|Heat conductivity
|<math>a_4</math>
|[W/m K5]
|<math>1.3852815 \cdot 10^{-10}</math>
|}

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Specification of <tt>NonlinearHeatCond-n410-nonlinear.zip</tt>.''
|-
|Property
|Symbol
|Unit
|Value
|-
|Number of nodes
|
|
|<math>410</math>
|-
|Beam length
|<math>l</math>
|[m]
|<math>0.1</math>
|-
|Cross-sectional area
|<math>A</math>
|[m2]
|<math>10^{-4}</math>
|-
|Material density
|<math>\rho</math>
|[kg/m3]
|<math>3970</math>
|-
|Heat capacity
|<math>C_p</math>
|[J/kg K]
|<math>766</math>
|-
|Heat conductivity
|<math>a_0</math>
|[W/m K]
|<math>36</math>
|-
|Heat conductivity
|<math>a_1</math>
|[W/m K2]
|<math>-0.1116</math>
|-
|Heat conductivity
|<math>a_2</math>
|[W/m K3]
|<math>0.00017298</math>
|-
|Heat conductivity
|<math>a_3</math>
|[W/m K4]
|<math>-1.78746 \cdot 10^{-7}</math>
|-
|Heat conductivity
|<math>a_4</math>
|[W/m K5]
|<math>1.3852815 \cdot 10^{-10}</math>
|}

The <tt>.m</tt> files contain matrices <math>E</math>, <math>A</math>, <math>B</math>, <math>F</math> and <math>C</math> and the vector <math>f</math> for the following system of equations:

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

The two outputs are on the left end and in the middle of the beam.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38883, see <ref name="lienemann05"/>.

==Data==

Details of the implementation are available in a separate report<ref name="lienemann04"/>.
A typical input to this system is a step response; periodic on/off switching is also possible.
The reduced model should thus both represent the step response as well as the possible influence of higher order harmonics.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/NonlinearHeatTransfer-dim1e1-n15-linear.zip NonlinearHeatTransfer-dim1e1-n15-linear.zip]
|1 kB
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/NonlinearHeatTransfer-dim1e1-n15-nonlinear.zip NonlinearHeatTransfer-dim1e1-n15-nonlinear.zip]
|1.2 kB
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/NonlinearHeatTransfer-dim1e2-n410-nonlinear.zip NonlinearHeatTransfer-dim1e2-n410-nonlinear.zip]
|18.8 kB
|}

==Dimensions==

System structure:

:<math>
\begin{align}
E\dot{x}(t) &= Ax(t) + Bu(t) + F f(x(t)) \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times 2}</math>,
<math>C \in \mathbb{R}^{2 \times N}</math>,
<math>E \in \mathbb{R}^{N \times N}</math>,
<math>F \in \mathbb{R}^{N \times N}</math>,
<math>f : \mathbb{R}^N \to \mathbb{R}^N</math>.

System variants:

<tt>n15-linear</tt>: <math>N = 15</math>,
<tt>n15-nonlinear</tt>: <math>N = 15</math>,
<tt>n410-nonlinear</tt>: <math>N = 410</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Nonlinear Heat Transfer'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Nonlinear_Heat_Transfer

@MISC{morwiki_nheat,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Nonlinear Heat Transfer},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Nonlinear_Heat_Transfer}</nowiki>,
year = {20XX}
}

* For the background on the benchmark:

@INPROCEEDINGS{LieYK04,
author = <nowiki>{J. Lienemann, A. Yousefi, J.G. Korvink}</nowiki>,
title = {Nonlinear heat transfer modelling},
booktile = {12th Mediterranean Conference on Control and Automation},
year = {2004}
}

==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="lienemann05"> J. Lienemann, A. Yousefi, J.G. Korvink, [https://doi.org/10.1007/3-540-27909-1_13 Nonlinear Heat Transfer Modeling], In: Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 327--331, 2005.</ref>

<ref name="lienemann04"> J. Lienemann, A. Yousefi, J.G. Korvink, [https://www.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/NonlinearHeatTransfer.pdf Nonlinear heat transfer modelling], 12th Mediterranean Conference on Control and Automation, 2004.</ref>

</references>

Linear 1D Beam

2023-11-30T09:33:38Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:ODE]]
[[Category:Linear]]
[[Category:Time invariant]]
[[Category:Second differential order]]
[[Category:SISO]]
[[Category:Sparse]]

{{Infobox
|Title = Linear 1D Beam
|Benchmark ID =
* linear1DBeam_n14m1q1
* linear1DBeam_n18m1q1
* linear1DBeam_n19994m1q1
* linear1DBeam_n19998m1q1
|Category = oberwolfach
|System-Class = LTI-SOS
|nstates =
* 14
* 18
* 19994
* 19998
|ninputs = 1
|noutputs = 1
|nparameters = 0
|components = B, C, E, K, M
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
* [[User:Mlinaric]]
* [[User:Hund]]
|Zenodo-link = NA
}}

==Description==
<figure id="fig1">[[File:Beam1.gif|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Beam2.gif|490px|thumb|right|Figure 2]]</figure>
<figure id="fig3">[[File:Beam3.gif|490px|thumb|right|Figure 3]]</figure>
<figure id="fig4">[[File:Beam4.gif|490px|thumb|right|Figure 4]]</figure>

Moving structures are an essential part of many microsystem devices, among them fluidic components like pumps and electrically controllable valves, sensing cantilevers, and optical structures.

Several actuation principles can be employed on microscopic length scales, the most frequent certainly the electromagnetic forces.
While electrostatic actuation falls behind at the macro scale, the effect of charged bodies outperforms magnetic forces in the micro scale both in terms of performance and fabrication expense.

While the single component can easily be simulated on a usual desktop computer, the calculation of a system of many coupled devices still presents a challenge.
This challenge is raised by the fact that many of these devices show a nonlinear behavior.
Especially for electrostatic structures, a further difficulty is the large reach of the electrostatic forces, leading to a strong spatial coupling of charges.

Accurate modelling of such a system typically leads to high order models.
The tasks of simulation, analysis and controller design of high order nonlinear control systems can be simplified by reducing the order of the original system and approximating it by a lower order model.

A application of electrostatic moving structures are e.g. [[wikipedia:RF_switch|RF switches]] or filters.
Given a simple enough shape, they often can be modelled as one-dimensional beams embedded in two or three dimensional space.

===Model Description===

This model describes a slender beam with four degrees of freedom per node:

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|<math>x</math>
|Axial displacement
|-
|<math>\theta_x</math>
|Axial rotation
|-
|<math>y</math>
|Flexural displacement
|-
|<math>\theta_z</math>
|Flexural rotation
|}

See Fig. 2 for Degree of Freedom <math>x</math>, Fig. 3 for Degree of Freedom <math>\theta_x</math> and Fig. 4 for Degrees of freedom <math>y</math> and <math>\theta_z</math>.

The beam is supported either on the left side or on both sides. For the left side (fixed) support,
the force is applied on the rightmost node in <math>y</math> direction, whereas for the support on both sides (simply supported), a node in the middle is loaded.
The damping matrix is calculated by a linear combination of the mass matrix <math>M</math> and the stiffness matrix <math>K</math>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38861.

==Data==

Based on the finite element discretization presented in<ref name="Weaver1990"/>, an interactive matrix generator has been created using [http://www.wolfram.com/ Wolfram Research]'s [http://www.wolfram.com/products/webmathematica webMathematica].
However, models produced by this generator are in the <tt>DSIF</tt><ref name="lienemsann2005"/> format, which allows for nonlinear terms.
For the purpose of the benchmark collection, we have precomputed four systems and converted them to the [http://math.nist.gov/MatrixMarket/ Matrix market] format which is easier to import in standard computer algebra packages.

All examples are made for a steel beam with the following properties:

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Property
|Value
|-
|Beam length (l)
|<math>0.1</math> m
|-
|Material density (rho)
|<math>8000</math> kg/m3
|-
|Cross-sectional area (A)
|<math>7.854\cdot 10^{-7}</math> m2
|-
|Moment of inertia (I)
|<math>4.909\cdot 10^{-14}</math> m4
|-
|Polar moment of inertia (J)
|<math>9.817\cdot 10^{-14}</math>
|-
|Modulus of elasticity (E)
|<math>2\cdot 10^{11}</math> Pa
|-
|Poisson ratio (nu)
|<math>0.29</math>
|-
|Contribution of <math>M</math> to damping
|<math>100</math>
|-
|Contribution of <math>K</math> to damping
|<math>0.01</math>
|-
|Support
|Simple, both sides
|}

The following examples are available (all files are compressed <tt>.zip</tt> archives, Units: SI):

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|File
|Degrees of freedom
|Number of nodes
|Number of equations
|File size [B]
|Compressed size [B]
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Linear1dBeam-dim1e1-LF10.zip Linear1dBeam-dim1e1-LF10.zip]
|flexural (<math>y</math> and <math>\theta_z</math>)
|10
|18
|5935
|2384
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Linear1dBeam-dim1e4-LF10000.zip Linear1dBeam-dim1e4-LF10000.zip]
|flexural (<math>y</math> and <math>\theta_z</math>)
|10000
|19998
|6640324
|716807
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Linear1dBeam-dim1e1-LFAT5.zip Linear1dBeam-dim1e1-LFAT5.zip]
|flexural (<math>y</math> and <math>\theta_z</math>), axial, torsional
|5
|14
|4045
|2255
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Linear1dBeam-dim1e5-LFAT5000.zip Linear1dBeam-dim1e5-LFAT5000.zip]
|flexural (<math>y</math> and <math>\theta_z</math>), axial, torsional
|5000
|19994
|5532532
|627991
|}

The zip files contain matrices <math>M</math>, <math>E</math>, <math>K</math>, <math>B</math> and <math>C</math> for the following system of equations:

:<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>

where <math>B</math> is a <math>n \times 1</math> matrix and <math>C</math> is a <math>1 \times n</math> matrix with the only nonzero entry at the <math>y</math> DOF of the middle node.

Details of the implementation are available in a separate [https://www.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Linear1dBeam.pdf report]. A typical input to this system is a step response; periodic on/off switching is also possible.
The reduced model should thus both represent the step response as well as the possible influence of higher order harmonics.
See also <ref name="lienemann2006"/>.

==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>

System variants:

<tt>LF10</tt>: <math>N = 18</math>,
<tt>LF100</tt>: <math>N = 19998</math>,
<tt>LFAT5</tt>: <math>N = 14</math>,
<tt>LFAT5000</tt>: <math>N = 19994</math>,

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

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection, '''Linear 1D Beam'''. hosted at MORwiki - Model Order Reduction Wiki, 2004. http://modelreduction.org/index.php/Linear_1D_Beam

@MISC{morwiki_linear_beam,
author = <nowiki>{{Oberwolfach Benchmark Collection}}</nowiki>,
title = {Linear 1{D} Beam},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Linear_1D_Beam}</nowiki>,
year = 2004
}

* For the background on the benchmark:

@TechReport{morLieRK06,
title = {MST MEMS Model Order Reduction: Requirements and Benchmarks},
author = {J. Lienemann, E.B. Rudnyi and J.G. Korvink},
journal = {Linear Algebra and its Applications},
year = 2006,
volume = 415,
issue = 2--3,
pages = {469--498},
month = {June},
publisher = {Elsevier},
doi = {10.1016/j.laa.2005.04.002}
}

==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="lienemsann2005"> J. Lienemann, B. Salimbahrami, B. Lohmann, J.G. Korvink, [https://doi.org/10.1007/3-540-27909-1_12 A File Format for the Exchange of Nonlinear Dynamical ODE Systems], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 317--326, 2005.</ref>

<ref name="Weaver1990"> W. Weaver Jr., S.P. Timoshenko, D.H. Young, [https://books.google.de/books?id=YZ7t8LgRqi0C&lpg=PP1&dq=editions%3AgXRPzjrsO3IC&pg=PP1#v=onepage&q&f=false Vibration problems in engineering], 5th ed., Wiley, 1990.</ref>

<ref name="lienemann2006"> J. Lienemann, E.B. Rudnyi, J.G. Korvink [https://doi.org/10.1016/j.laa.2005.04.002 MST MEMS model order reduction: Requirements and benchmarks], Linear Algebra and its Applications 415(2--3): 469--498, 2006.</ref>

</references>

International Space Station

2023-11-30T09:32:55Z

Malhotra: infobox string

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

{{Infobox
|Title = International Space Station
|Benchmark ID =
* iss_n1412m3q3
* iss_n270m3q3
|Category = slicot
|System-Class = LTI-FOS
|nstates =
* 1412
* 270
|ninputs = 3
|noutputs = 3
|nparameters = 0
|components = A, B, C
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
|Zenodo-link = NA
}}

==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>

Heat Equation

2023-11-30T09:32:27Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:Sparse]]
[[Category:SISO]]

{{Infobox
|Title = Heat Equation
|Benchmark ID = heatEquation_n200m1q1
|Category = slicot
|System-Class = LTI-FOS
|nstates = 200
|ninputs = 1
|noutputs = 1
|nparameters = 0
|components = A, B, C, E
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
|Zenodo-link = NA
}}

==Description: Discretization of the Heat Equation==

This benchmark represents a discretized one-dimensional [[wikipedia:Heat_equation|heat equation]] model.
More details can be found in <ref name="chahlaoui02"/>.

For a large-scale heat equation example see the [[Steel_Profile|steel profile]] benchmark.

==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>, <math>E</math> 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/heat-disc.zip heat-disc.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

==Dimensions==

System structure:

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

System dimensions:

<math>A \in \mathbb{R}^{200 \times 200}</math>,
<math>B \in \mathbb{R}^{200 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 200}</math>,
<math>E \in \mathbb{R}^{200 \times 200}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MANUAL{slicot_heat,
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 background on the benchmark:

@TECHREPORT{morChaV92,
author = <nowiki>{Y. Chahlaoui, P. Van Dooren}</nowiki>,
title = {A collection of Benchmark examples for model reduction of linear time invariant dynamical systems},
institution = {SLICOT Working Note},
number = {2002-2},
year = {2002},
url = <nowiki>{http://eprints.maths.manchester.ac.uk/1040/1/ChahlaouiV02a.pdf}</nowiki>
}

==References==

<references>

<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>

Gas Sensor

2023-11-30T09:32:08Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:linear]]
[[Category:ODE]]
[[Category:first differential order]]
[[Category:time invariant]]

{{Infobox
|Title = Gas Sensor
|Benchmark ID = gasSensor_n66917m1q28
|Category = oberwolfach
|System-Class = LTI-FOS
|nstates = 66917
|ninputs = 1
|noutputs = 28
|nparameters = 0
|components = A, B, C, E
|License = NA
|Creator = [[User:Feng]]
|Editor =
* [[User:Feng]]
* [[User:Himpe]]
* [[User:Will]]
|Zenodo-link = NA
}}

==Description: Microhotplate Gas Sensor==
<figure id="fig1">[[File:GasSensor1.jpg|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:GasSensor2.jpg|490px|thumb|right|<caption>Masks disposition (left) and the schematical position of the chosen output nodes (right).</caption>]]</figure>

The goal of European project [https://web.archive.org/web/20050826075758/http://www.cnm.es:80/imb/glassgas/index.htm Glassgas] (IST-99-19003) was to develop a novel metal oxide low power '''microhotplate gas sensor''' <ref name="wollenstein03"/>.
In order to assure a robust design and good thermal isolation of the membrane from the surrounding wafer, the silicon microhotplate is supported by glass pillars emanating from a glass cap above the silicon wafer, as shown in Fig. 1.
In this design, four different sensitive layers can be deposited on the membrane.
The thermal management of a '''microhotplate gas sensor''' is of crucial importance.

The benchmark contains a thermal model of a single gas sensor device with three main components:
a silicon rim, a silicon hotplate and glass structure <ref name="hildenbrand03"/>.
It allows us to simulate important thermal issues, such as the homogeneous temperature distribution over gas sensitive regions or thermal decoupling between the hotplate and the silicon rim.
The original model is the heat transfer partial differential equation.

The device solid model has been made and then meshed and discretized in [http://www.ansys.com ANSYS] 6.1 by means of the finite element method (<tt>SOLID70</tt> elements were used).
It contains 68000 elements and 73955 nodes.
Material properties were considered as temperature independent.
Temperature is assumed to be in degree Celsius with the initial state of <math>0 C</math>.
The Dirichlet boundary conditions of <math>T = 0 C</math> is applied at the top and bottom of the chip (at 7038 nodes).

The output nodes are described in Table 1.
In Fig. 2 the red marked nodes are positioned on the silicon rim.
Their temperature should be close to the initial temperature in the case of good thermal decoupling between the membrane and the silicon rim.
The black marked nodes are placed on the sensitive layers above the heater and are numbered from left to right row by row, as schematically shown in Fig 2.
They allow us to prove whether the temperature distribution over the gas sensitive layers is homogeneous (maximum difference of <math>10C</math> is allowed by design).

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Inputs and outputs for the gas sensor model.''
|-
|Number
|Code
|Comment
|-
|1
|aHeater
|within a heater, to be used for nonlinear input
|-
|2-7
|SiRim1 to SiRim7
|silicon rim
|-
|8-28
|Memb1 to Memb21
|gas sensitive layer
|}

The benchmark contains a constant load vector.
The input function equal to <math>u(t) = 1</math> corresponds to the constant input power of <math>340 mW</math>.
One can insert a weak input nonlinearity related to the dependence of heater's resistivity on temperature given as:

:<math>
R(T) = R_{0}(1 + \alpha T)
</math>

where <math>\alpha =1.469 \cdot 10^{-3} K^{-1}</math>.
To this end, one has to multiply the load vector by a function:

:<math>
\frac{U^2 274.94 (1 + \alpha T)}{0.34 (274.94 (1 + \alpha T)+148.13)^2}
</math>

where <math>U</math> is a desired constant voltage.
The temperature in the equation above should be replaced by the temperature at the input 1 (aHeater).

The linear ordinary differential equations of the first order are written as:

:<math>
\begin{array}{rcl}
E \frac{\partial T}{\partial t} &=& A T(t) + B u(t) \\
y(t) &=& C T(t)
\end{array}
</math>

where <math>E</math> and <math>A</math> are the symmetric sparse system matrices (heat capacity and heat conductivity matrix), <math>B</math> is the load vector, <math>C</math> is the output matrix, and <math>T</math> is the vector of unknown temperatures.
The dimension of the system is <math>66917</math>, the number of nonzero elements in matrix <math>E</math> is <math>66917</math>, in matrix <math>A</math> is <math>885141</math>.

The outputs of the transient simulation at output 18 (Memb11) over the rise time of the device of <math>5 s</math> for the original linear (with constant input power of <math>340 mW</math>) and nonlinear (with constant voltage of <math>14 V</math>) model are placed in files <tt>LinearResults</tt> and <tt>NonlinearResults</tt> respectively.
The results can be used to compare the solution of a reduced model with the original one.
The time integration has been performed in ANSYS with accuracy of about <math>0.1 \%</math>.
The results are given as matrices where the first row is made of times, the second of the temperatures.

More information can also be found in <ref name="bechthold05"/>.

==Data==

Download matrices in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format: [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/GasSensor-dim1e5-GasSensor.tar.gz GasSensor-dim1e5-GasSensor.tar.gz], (8 MB).

The matrix name is used as an extension of the matrix file.
File <tt>*.C</tt> names contains a list of output names written consecutively.
The system matrices have been extracted from ANSYS models by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem].

The discussion of electro-thermal modeling related to the benchmark including the nonlinear input function can be found in <ref name="bechthold04"/>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38880, see <ref name="hildenbrand2005"/>.

==Dimensions==

System structure:

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

System dimensions:

<math>E \in \mathbb{R}^{66917 \times 66917}</math>,
<math>A \in \mathbb{R}^{66917 \times 66917}</math>,
<math>B \in \mathbb{R}^{66917 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 66917}</math>.

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

@INPROCEEDINGS{BecHWetal04,
author = <nowiki>{T. Bechtold, J. Hildenbrand, J. Wöllenstein, J. G. Korvink}</nowiki>,
title = {Model Order Reduction of 3D Electro-Thermal Model for a Novel, Micromachined Hotplate Gas Sensor},
booktitle = {Proceedings of 5th International Conference on Thermal and Mechanical Simulation and Experiments in Microelectronics and Microsystems},
pages = {263--267},
year = {2004},
doi = {10.1109/ESIME.2004.1304049}
}

==References==

<references>

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

<ref name="hildenbrand2005">Hildenbrand J., Bechtold T., J. Wöllenstein, [https://doi.org/10.1007/3-540-27909-1_14 Microhotplate Gas Sensor]. In: Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol 45: 333-336, 2005.</ref>

<ref name="wollenstein03">J. Wöllenstein, H. Böttner, J.A. Pláza, C. Carné, Y. Min, H.L. Tuller, [https://doi.org/10.1016/S0925-4005(03)00218-1 A novel single chip thin film metal oxide array], Sensors and Actuators B: Chemical 93 (1-3): 350--355, 2003.</ref>

<ref name="hildenbrand03">J. Hildenbrand, [https://www.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/GasSensor-Hildenbrand03.pdf Simulation and Characterisation of a Gas sensor and Preparation for Model Order Reduction], Diploma Thesis, University of Freiburg, Germany, 2003.</ref>

<ref name="bechthold04">T. Bechtold, J. Hildenbrand, J. Wöllenstein, J. G. Korvink, [https://doi.org/10.1109/ESIME.2004.1304049 Model Order Reduction of 3D Electro-Thermal Model for a Novel, Micromachined Hotplate Gas Sensor], Proceedings of 5th International Conference on Thermal and Mechanical Simulation and Experiments in Microelectronics and Microsystems, EUROSIME2004, May 10-12, 2004, Brussels, Belgium: 263--267, 2004.</ref>

<ref name="bechthold05">T. Bechtold, "[https://www.freidok.uni-freiburg.de/volltexte/1914/ Model Order Reduction of Electro-Thermal MEMS]", PhD thesis, Department of Microsystems Engineering, University of Freiburg, 2005.</ref>

</references>

==Contact==

'' [[User:Feng|Lihong Feng]] ''

[https://www.jade-hs.de/team/tamara-bechtold/ Tamara Bechtold]

Electrostatic Beam

2023-11-30T09:31:12Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Oberwolfach]]

{{Infobox
|Title = Electrostatic Beam
|Benchmark ID =
* electrostaticBeam_n38m1q1
* electrostaticBeam_n398m1q1
|Category = oberwolfach
|System-Class = NLTI-SOS
|nstates =
* 38
* 398
|ninputs = 1
|noutputs = 1
|nparameters = 0
|components = B, C, E, F, K, M, f
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
|Zenodo-link = NA
}}

==Description: Beam Actuated by Electrostatic Force==

<figure id="fig1">[[File:EBeam.gif|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Beam4.gif|490px|thumb|right|<caption>Degrees of freedom <math>y</math> and <math>\theta_z</math></caption>]]</figure>

Moving structures are an essential part for many microsystem devices, among them fluidic components like pumps and electrically controllable valves, sensing cantilevers, and optical structures.

Several actuation principles can be employed on microscopic length scales, certainly most frequent electromagnetic forces.
While electrostatic actuation falls behind at the macro scale, the effect of charged bodies outperforms magnetic forces in the micro scale both in terms of performance and manufacturing expense.

While the single component can easily be simulated on a usual desktop computer, the calculation of a system of many coupled devices still presents a challenge.
This challenge is raised by the fact that many of these devices show a nonlinear behavior.
Especially for electrostatic structures, a further difficulty is the large reach of the electrostatic forces, leading to a strong spatial coupling of charges.

Accurate modelling of such a system typically leads to high order models.
The tasks of simulation, analysis and controller design of high order nonlinear control systems can be simplified by reducing the order of the original system and approximating it by a lower order model.

An application of electrostatic moving structures are e.g. [[wikipedia:RF_switch|RF switches]] or filters.
Given a simple enough shape, they often can be modelled as one-dimensional beams embedded in two or three dimensional space.

===Model description===

This model describes a slender beam which is actuated by a voltage between the beam and the ground electrode below (see Fig. 1).

On the beam, at least three degrees of freedom per node have to be considered:

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|<math>y</math>
|Flexural displacement
|-
|<math>\theta_z</math>
|Flexural rotation
|-
|<math>q</math>
|Charge
|}

On the ground electrode, all spatial degrees of freedom are fixed, so only charge has to be considered.
The beam is supported either on the left side or on both sides.
The damping matrix is calculated by a linear combination of the mass matrix <math>M</math> and the stiffness matrix <math>K</math>.

The calculation of the electrostatic force would require a boundary element discretization, where it would be necessary to recalculate the capacity matrix for each time-step due to the motion of the charges
This would require an integration over the beam's elements and could be written in analytical form by using e.g. [[wikipedia:Gaussian_quadrature|Gauss integration]];
however, the complexity of the resulting system would be too high.
We therefore use the method shown in <ref name="siverberg1996"/>, i.e. we concentrate the charges on the nodes.
The capacity matrix then follows a simple <math>1/r</math> law.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>.

==Data==

Based on the finite element discretization presented in <ref name="weaver1990"/>, an interactive matrix generator has been created using [http://www.wolfram.com/ Wolfram Research]'s [http://www.wolfram.com/products/webmathematica webMathematica].
Models produced by this generator are in the DSIF<ref name="lienemsann2005"/> format, which allows for nonlinear terms.

All examples are made for a silicon beam with the following properties:

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
! Property
! Value
|-
|Beam length (l)
|<math>10^{-5} m</math>
|-
|Beam height (h)
|<math>10^{-6} m</math>
|-
|Beam width (w)
|<math>15 \cdot 10^{-6} m</math>
|-
|Distance between beams (s)
|<math>200 \cdot 10^{-9} m</math>
|-
|Material density (rho)
|<math>2330 kg/m^3</math>
|-
|Cross-sectional area (A)
|<math>150 \cdot 10^{12} m^2</math>
|-
|Moment of inertia (I)
|<math>1.25 \cdot 10^{-21} m^4</math>
|-
|Modulus of elasticity (E)
|<math>1.31 \cdot 10^{11} Pa</math>
|-
|Contribution of <math>M</math> to damping
|<math>0</math>
|-
|Contribution of <math>K</math> to damping
|<math>10^{-6}</math>
|-
|Support
|Both sides, <math>y</math> DOF only
|}

The following examples are available (all files are zipped compressed DSIF files, Units: SI):

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
! File
! Number of Nodes
! Number of Equations
! Compressed Size [kB]
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ElectrostaticBeam-dim1e1-E10.zip ElectrostaticBeam-dim1e1-E10.zip]
|10
|38
|4144
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ElectrostaticBeam-dim1e2-E100.zip ElectrostaticBeam-dim1e2-E100.zip]
|100
|398
|347679
|}

The <tt>.m</tt> files contain matrices <math>M</math>, <math>E</math>, <math>K</math>, <math>B</math>, <math>F</math> and <math>C</math>, the vector <math>f</math> and initial conditions for the following system of equations:

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

where <math>B</math> is a <math>n \times 1</math> matrix with <math>1</math> at all charge DOFs of the upper beam and <math>C</math> is a <math>1 \times n</math> matrix with the only nonzero entry at the <math>y</math> DOF of the middle node.

Details of the implementation are available in a separate report ([https://www.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ElectrostaticBeam.pdf ElectrostaticBeam.pdf]), see also <ref name="lienemann2006"/>.

A typical input to this system is a step response; periodic on/off switching is also possible.
The reduced model should thus both represent the step response as well as the possible influence of higher order harmonics.

==Dimensions==

System structure:

:<math>
\begin{align}
M \ddot{x}(t) + E\dot{x}(t) + K x(t) &= B u(t) + F f(x(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>F \in \mathbb{R}^{N \times S}</math>,
<math>f : \mathbb{R}^N \to \mathbb{R}^S</math>,
<math>C \in \mathbb{R}^{1 \times N}</math>.

System variants:
<tt>E10</tt>: <math>N = 38</math>, <math>S = 28</math>,
<tt>E100</tt>: <math>N = 398</math>, <math>S = 298</math>.

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

@ARTICLE{morLieRK06,
author = <nowiki>{J. Lienemann and E.B. Rudnyi and J.G. Korvink}</nowiki>,
title = {{MST} {MEMS} model order reduction: Requirements and benchmarks},
journal = {Journal of Biomechanics},
volume = {415},
number = {2--3},
pages = {469--498},
year = {2006},
doi = {10.1016/j.laa.2005.04.002}
}

==References==

<references>

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

<ref name="lienemsann2005"> J. Lienemann, B. Salimbahrami, B. Lohmann, J.G. Korvink, [https://doi.org/10.1007/3-540-27909-1_12 A File Format for the Exchange of Nonlinear Dynamical ODE Systems], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 317--326, 2005.</ref>

<ref name="siverberg1996">L. Silverberg, L. Weaver, Jr., [https://doi.org/10.1115/1.2788876 Dynamics and Control of Electrostatic Structures], Journal of Applied Mechanics, Vol. 63, p. 383--391, 1996.</ref>

<ref name="weaver1990">W. Weaver, Jr., S.P. Timoshenko, D.H. Young, [https://books.google.de/books?id=YZ7t8LgRqi0C&lpg=PP1&dq=editions%3AgXRPzjrsO3IC&pg=PP1#v=onepage&q&f=false Vibration problems in engineering], 5th ed., Wiley, 1990.</ref>

<ref name="lienemann2006">J. Lienemann, E.B. Rudnyi, J.G. Korvink, [https://doi.org/10.1016/j.laa.2005.04.002 MST MEMS model order reduction: Requirements and benchmarks], Linear Algebra and its Applications, 415(2--3): 469--498 , 2006.</ref>

</references>

Earth Atmosphere

2023-11-30T09:30:51Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:SISO]]
[[Category:Dense]]

{{Infobox
|Title = Earth Atmosphere
|Benchmark ID = earthAtmosphere_n598m1q1
|Category = slicot
|System-Class = LTI-FOS
|nstates = 598
|ninputs = 1
|noutputs = 1
|nparameters = 0
|components = A, B, C
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
* [[User:Mlinaric]]
|Zenodo-link = NA
}}

==Description: Model of an Atmospheric Storm Track==

This benchmark models the track of an [[wikipedia:Storm_track|atmospheric storm track]].
More details can be found in <ref name="farrell95"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

==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> 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/eady.zip eady.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

Here is [https://www.python.org Python] code for loading the matrices (<math>B</math> and <math>C</math> are stored as arrays of 8-bit unsigned integers):

:<syntaxhighlight lang="python">
import numpy as np
from scipy.io import loadmat

mat = loadmat('eady.mat')
A = mat['A']
B = mat['B'].astype(np.float64)
C = mat['C'].astype(np.float64)
</syntaxhighlight>

==Dimensions==

System structure:

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

System dimensions:

<math>A \in \mathbb{R}^{598 \times 598}</math>,
<math>B \in \mathbb{R}^{598 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 598}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MANUAL{slicot_cdisc,
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 background on the benchmark:

@ARTICLE{FarI95,
author = <nowiki>{B.F. Farrell and P.J. Ioannou}</nowiki>,
title = {Stochastic Dynamics of the Midlatitude Atmospheric Jet},
journal = {Journal of the Atmospheric Sciences},
volume = {52},
number = {10},
pages = {1642--1656},
year = {1995},
doi = {10.1175/1520-0469(1995)052<1642:SDOTMA>2.0.CO;2}
}

==References==

<references>

<ref name="farrell95"> B.F. Farrell, P.J. Ioannou. [https://doi.org/10.1175/1520-0469(1995)052%3C1642:SDOTMA%3E2.0.CO;2 Stochastic Dynamics of the Midlatitude Atmospheric Jet]. Journal of the Atmospheric Sciences, 52(10): 1642--1656, 1995.</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>

Convective Thermal Flow

2023-11-30T09:29:32Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:MIMO]]
[[Category:sparse]]

{{Infobox
|Title = Convective Thermal Flow
|Benchmark ID =
* ctfChipCoolingv0_n20082m1q5
* ctfChipCoolingv01_n20082m1q5
* ctfFlowMeterv0_n9669m1q5
* ctfFlowMeterv05_n9669m1q5
|Category = oberwolfach
|System-Class = LTI-FOS
|nstates =
* 20082
* 20082
* 9669
* 9669
|ninputs = 1
|noutputs = 5
|nparameters = 0
|components = A, B, C
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
|Zenodo-link = NA
}}

==Description: Convective Thermal Flow Problems==
<figure id="fig1">[[File:Convection1.jpg|490px|thumb|right|Figure 1: Convective heat flow example: 2D anemometer model]]</figure>
<figure id="fig2">[[File:Convection2.jpg|490px|thumb|right|Figure 2: Convective heat flow example: 3D cooling structure]]</figure>

Many thermal problems require simulation of heat exchange between a solid body and a fluid flow.
The most elaborate approach to this problem is [[wikipedia:Computational_fluid_dynamics|computational fluid dynamics]] (CFD).
However, CFD is computationally expensive.
A popular solution is to exclude the flow completely from the computational domain and to use convection boundary conditions for the solid model.
However, caution has to be taken to select the [[wikipedia:Heat_transfer_coefficient|film coefficient]].

An intermediate level is to include a flow region with a given velocity profile that adds convective transport to the model.
Compared to convection boundary conditions this approach has the advantage that the film coefficient does not need to be specified and that information about the heat profile in the flow can be obtained.
A drawback of the method is the greatly increased number of elements needed to perform a physically valid simulation because the solution accuracy when employing upwind finite element schemes depends on the element size.
While this problem still is linear, due to the forced convection, the conductivity matrix changes from a symmetric matrix to an un-symmetric one.
So this problem type can be used as a benchmark for problems containing un-symmetric matrices.

Two different designs are tested: a 2D model of an [[Anemometer|anemometer]]-like structure mainly consisting of a tube and a small heat source (Fig. 1) <ref name="ernst2001"/>.
The solid model has been generated and meshed in [http://www.ansys.com ANSYS].
Triangular <tt>PLANE55</tt> elements have been used for meshing and discretizing by the finite element method, resulting in 19282 elements and 9710 nodes.
The second design is a 3D model of a chip cooled by forced convection (Fig. 2) <ref name="harper1997"/>.
In this case, the tetrahedral element type <tt>SOLID70</tt> was used, resulting in 107989 elements and 20542 nodes.
Since the implementation of the convective term in ANSYS does not allow defining the fluid speed on a per-element basis but on a per-region one, the flow profile has to be approximated by piece-wise step functions.
The approximation used for these benchmarks is shown in Fig. 1.

The Dirichlet boundary conditions are applied to the original system.
In both models, the reference temperature is set to <math>300 \, \text{K}</math>, and Dirichlet boundary conditions as well as initial conditions are set to <math>0</math> with respect to the reference.
The specified Dirichlet boundary conditions are in both cases the inlet of the fluid and the outer faces of the solids. Matrices are supplied for the symmetric case (fluid speed is zero; no convection), and the non-symmetric case (with forced convection).
Table 1 shows the output nodes specified for the two benchmarks, Table 2 links the filenames according to the different cases.

Practically, only a few nodes are considered quantities of interest.
Hence, a small subset of five nodes is selected as output nodes,
which are filtered from the discretized state by a linear transformation.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38867, see <ref name="moosmann2005"/>.

==Data==

Matrices are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format.
The matrix name is used as an extension of the matrix file.
<tt>*.C.names</tt> contains a list of output names written consecutively.
The system matrices have been extracted from ANSYS models by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem].

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Output nodes for the two models.''
|-
!Model
!Number
!Code
!Comment
|-
|Flow Meter
|1
|out1
|outlet position
|-
|
|2
|out2
|outlet position
|-
|
|3
|SenL
|left sensor position
|-
|
|4
|Heater
|within the heater
|-
|
|5
|SenR
|right sensor position
|-
|Cooling Structure
|1
|out1
|outlet position
|-
|
|2
|out2
|outlet position
|-
|
|3
|out3
|outlet position
|-
|
|4
|out4
|outlet position
|-
|
|5
|Heater
|within the heater
|}

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 2: Provided files.''
|-
!Model
!Fluid Speed (m/s)
!Link
!Size
|-
|Flow Meter
|<math>0</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-flow_meter_model_v0.tgz Convection-dim1e4-flow_meter_model_v0.tgz]
|649.4 kB
|-
|
|<math>0.5</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-flow_meter_model_v0.5.tgz Convection-dim1e4-flow_meter_model_v0.5.tgz]
|757.8 kB
|-
|Cooling Structure
|<math>0</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-chip_cooling_model_v0.tgz Convection-dim1e4-chip_cooling_model_v0.tgz]
|3.9 MB
|-
|
|<math>0.1</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-chip_cooling_model_v0.1.tgz Convection-dim1e4-chip_cooling_model_v0.1.tgz]
|4.0 MB
|}

Further information on the models can be found in <ref name="moosmann2004"/>,
where model reduction by means of the [[wikipedia:Arnoldi_iteration|Arnoldi algorithm]] is also presented.

==Dimensions==

System 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 1}</math>,
<math>C \in \mathbb{R}^{5 \times n}</math>.

System variants:

<tt>flow_v0</tt>: <math>n = 9669</math>,
<tt>flow_v0.5</tt>: <math>n = 9669</math>,
<tt>chip_v0</tt>: <math>n = 20082</math>,
<tt>chip_v0.1</tt>: <math>n = 20082</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Convective Thermal Flow'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Convective_Thermal_Flow

@MISC{morwiki_convection,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Convective Thermal Flow},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Convection}</nowiki>,
year = {20XX}
}

* For the background on the benchmark:

@INPROCEEDINGS{morMooRGetal04,
author = <nowiki>{C. Moosmann, E.B. Rudnyi, A. Greiner, J.G. Korvink}</nowiki>,
title = {Model Order Reduction for Linear Convective Thermal Flow},
booktitle = {Proceedings of 10th International Workshops on THERMal INvestigations of ICs and Systems},
year = {2004},
url = <nowiki>{http://modelreduction.com/doc/papers/moosmann04THERMINIC.pdf}</nowiki>
}

==References==

<references>

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

<ref name="ernst2001">H. Ernst, [https://freidok.uni-freiburg.de/data/201 High-Resolution Thermal Measurements in Fluids], PhD thesis, University of Freiburg, Germany, 2001.</ref>

<ref name="harper1997">C.A. Harper, [https://doi.org/10.1036/0071430482 Electronic packaging and interconnection handbook], New York McGraw- Hill, USA, 1997</ref>

<ref name="moosmann2004">C. Moosmann, E.B. Rudnyi, A. Greiner, J.G. Korvink, [http://modelreduction.com/doc/papers/moosmann04THERMINIC.pdf Model Order Reduction for Linear Convective Thermal Flow], Proceedings of 10th International Workshops on THERMal INvestigations of ICs and Systems, THERMINIC2004, Sophia Antipolis, France, 2004.</ref>

<ref name="moosmann2005">C. Moosmann, A. Greiner, [https://doi.org/10.1007/3-540-27909-1_16 Convective Thermal Flow Problems], In: Dimension Reduction of Large-Scale Systems. Springer, Berlin, Heidelberg. Lecture Notes in Computational Science and Engineering, vol 45: 341--343, 2005.</ref>

</references>

Convection Reaction

2023-11-30T09:28:49Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:Sparse]]
[[Category:SISO]]

{{Infobox
|Title = Convection Reaction
|Benchmark ID = convectionReaction_n84m1q1
|Category = slicot
|System-Class = LTI-FOS
|nstates = 84
|ninputs = 1
|noutputs = 1
|nparameters = 0
|components = A, B, C
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
* [[User:Mlinaric]]
|Zenodo-link = NA
}}

==Description==

This benchmark models a chemical reaction by a [[wikipedia:Convection-diffusion_equation|convection]]-[[wikipedia:Reaction-diffusion|reaction]] partial differential equation on the unit square,
given by:

:<math>
\frac{\partial x}{\partial t} = \frac{\partial^2 x}{\partial y^2} + \frac{\partial^2 x}{\partial z^2} + 20 \frac{\partial x}{\partial z} - 180 x + f(y,z) x(t),
</math>

with Dirichlet boundary conditions and discretized with centered difference approximation.

The input vector <math>B</math> is composed of random elements and the output vector equals the input vector <math>C = B^T</math>.

More details can be found in <ref name="raschman80"/>, <ref name="saad88"/>, <ref name="grimme97"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

==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> are available from the [https://www.slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [https://www.slicot.org/objects/software/shared/bench-data/pde.zip pde.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

Here is [https://www.python.org Python] code for loading the matrices (<math>A</math> is a sparse matrix of 16-bit integers and <math>B</math> and <math>C</math> are full matrices stored as sparse matrices):

:<syntaxhighlight lang="python">
import numpy as np
from scipy.io import loadmat

mat = loadmat('build.mat')
A = mat['A'].astype(np.float64)
B = mat['B'].toarray()
C = mat['C'].toarray()
</syntaxhighlight>

==Dimensions==

System structure:

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

System dimensions:

<math>A \in \mathbb{R}^{84 \times 84}</math>,
<math>B \in \mathbb{R}^{84 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 84}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MANUAL{slicot_pde,
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 background on the benchmark:

@ARTICLE{Saa88,
author = <nowiki>{Y. Saad}</nowiki>,
title = {Projection and deflation method for partial pole assignment in linear state feedback},
journal = {IEEE Transactions on Automatic Control},
volume = {33},
number = {3},
pages = {290--297},
year = {1988},
doi = {10.1109/9.406}
}

==References==

<references>

<ref name="raschman80"> P. Raschman, M. Kuhicek, M. Maros. Waves in distributed chemical systems: Experiments and computations. In: New Approaches to Nonlinear Problems in Dynamics - Proceedings of the Asilomar Conference Ground: 271--288, SIAM, 1980.</ref>

<ref name="saad88"> Y. Saad. [https://doi.org/10.1109/9.406 Projection and deflation method for partial pole assignment in linear state feedback], IEEE Transactions on Automatic Control, 33(3): 290--297, 1988.</ref>

<ref name="grimme97"> E.J. Grimme. [https://www.proquest.com/dissertations-theses/krylov-projection-methods-model-reduction/docview/304361372/se-2?accountid=14597 Krylov Projection Methods for Model Reduction]. PhD Thesis, University of Illinois at Urbana-Champaign, 1998.</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>

Clamped Beam

2023-11-30T09:28:21Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:second differential order]]
[[Category:SISO]]

{{Infobox
|Title = Clamped Beam
|Benchmark ID = clampedBeam_n348m1q1
|Category = slicot
|System-Class = LTI-FOS
|nstates = 348
|ninputs = 1
|noutputs = 1
|nparameters = 0
|components = A, B, C
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
* [[User:Mlinaric]]
|Zenodo-link = NA
}}

==Description: Clamped Beam Model==

This benchmark models a [[wikipedia:Cantilever|cantilever beam]], which is a beam clamped on one end.
More details can be found in <ref name="antoulas01"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

For larger beam-type benchmarks see the [[Linear_1D_Beam|linear 1d beam]] and [[Electrostatic_Beam|electrostatic beam]] benchmarks.

==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> are available from the [https://www.slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [https://www.slicot.org/objects/software/shared/bench-data/beam.zip beam.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

Here is [https://www.python.org Python] code for loading the matrices (<math>A</math> is stored as a sparse matrix that is mostly full and <math>C</math> is stored as an array of 8-bit unsigned integers):

:<syntaxhighlight lang="python">
import numpy as np
from scipy.io import loadmat

mat = loadmat('beam.mat')
A = mat['A'].toarray()
B = mat['B']
C = mat['C'].astype(np.float64)
</syntaxhighlight>

The <math>(A, B, C)</math> represents a second-order system

:<math>
\begin{align}
\ddot{q}(t) + E_{so} \dot{q}(t) + a K_{so} q(t) &= a B_{so} u(t), \\
y(t) &= C_{so} q(t),
\end{align}
</math>

as

:<math>
\begin{align}
A &=
\begin{pmatrix}
0 & a I \\
-K_{so} & -E_{so}
\end{pmatrix}, \\
B &=
\begin{pmatrix}
0 \\
B_{so}
\end{pmatrix}, \\
C &=
\begin{pmatrix}
C_{so} & 0
\end{pmatrix},
\end{align}
</math>

where <math>a \approx 21.3896</math>.

Here is [https://www.python.org Python] code for checking the structure and extracting the second-order matrices:

:<syntaxhighlight lang="python">
n = 348
n2 = n // 2

assert np.all(A[:n2, :n2] == 0)
assert np.all(A[:n2, n2:] == A[0, n2] * np.eye(n2))
assert np.all(B[:n2] == 0)
assert np.all(C[:, :n2] == 0)

a = A[0, n2]
Eso = -A[n2:, n2:]
Kso = -a * A[n2:, :n2]
Bso = a * B[n2:]
Cso = C[:, :n2]
</syntaxhighlight>

==Dimensions==

===First differential order===

System structure:

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

System dimensions:

<math>A \in \mathbb{R}^{348 \times 348}</math>,
<math>B \in \mathbb{R}^{348 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 348}</math>.

===Second differential order===

System structure:

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

System dimensions:

<math>E, K \in \mathbb{R}^{174 \times 174}</math>,
<math>B \in \mathbb{R}^{174 \times 1}</math>,
<math>C_p \in \mathbb{R}^{1 \times 174}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MANUAL{slicot_beam,
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 background on the benchmark:

@ARTICLE{morAntSG01,
author = <nowiki>{A.C. Antoulas, D.C. Sorensen and S. Gugercin}</nowiki>,
title = {A survey of model reduction methods for large-scale systems},
journal = {Contemporary Mathematics},
volume = {280},
pages = {193--219},
year = {2001},
doi = {10.1090/conm/280}
}

==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="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>

Circular Piston

2023-11-30T09:27:49Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:second differential order]]
[[Category:Sparse]]
[[Category:MIMO]]

{{Infobox
|Title = Circular Piston
|Benchmark ID = circularPiston_n2025m1q2025
|Category = oberwolfach
|System-Class = LTI-SOS
|nstates = 2025
|ninputs = 1
|noutputs = 2025
|nparameters = 0
|components = B, C, E, K, M
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
* [[User:Mlinaric]]
|Zenodo-link = NA
}}

==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) + K x(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> and the output is the state vector <math>x</math>.
The motivation for using model reduction for this type of problem 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).

Extracting will produce

* <tt>piston.M</tt>
* <tt>piston.E</tt>
* <tt>piston.K</tt>
* <tt>piston.B</tt>

Note that for <tt>piston.B</tt>, loading with <tt>scipy.io.mmread</tt> will not work because the number of nonzeros is specified in the dense Matrix Market format.
Replacing the line with "2025 1 2025" by "2025 1" will make it work.

==Dimensions==

System structure:

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

System dimensions:

<math>M, E, 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>

CD Player

2023-11-30T09:27:23Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:Sparse]]
[[Category:MIMO]]

{{Infobox
|Title = CD Player
|Benchmark ID = cdPlayer_n120m2q2
|Category = slicot
|System-Class = LTI-FOS
|nstates = 120
|ninputs = 2
|noutputs = 2
|nparameters = 0
|components = A, B, C
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
|Zenodo-link = NA
}}

==Description: Classical CD Player Model==

This benchmark models the swing arm of a [[wikipedia:CD_Player|CD Player]] holding a lens which can be moved in the horizontal plain.
More details can be found in <ref name="draijer92"/>, <ref name="wortelboer96"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

==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> are available from the [https://www.slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [https://www.slicot.org/objects/software/shared/bench-data/CDplayer.zip CDplayer.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

==Dimensions==

System structure:

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

System dimensions:

<math>A \in \mathbb{R}^{120 \times 120}</math>,
<math>B \in \mathbb{R}^{120 \times 2}</math>,
<math>C \in \mathbb{R}^{2 \times 120}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MANUAL{slicot_cdisc,
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 background on the benchmark:

@ARTICLE{DraSB92,
author = <nowiki>{W. Draijer and M. Steinbuch and O. Bosgra}</nowiki>,
title = {Adaptive Control of the Radial Servo System of a Compact Disc Player},
journal = {Automatica},
volume = {28},
number = {3},
pages = {455--462},
year = {1992},
doi = {10.1016/0005-1098(92)90171-B}
}

==References==

<references>

<ref name="draijer92"> W. Draijer, M. Steinbuch, O. Bosgra. [https://doi.org/10.1016/0005-1098(92)90171-B Adaptive Control of the Radial Servo System of a Compact Disc Player]. Automatica, 28(3): 455--462, 1992.</ref>

<ref name="wortelboer96"> P. Wortelboer, M. Steinbuch, O. Bosgra. [https://research.tue.nl/en/publications/closed-loop-balanced-reduction-with-cation-to-a-compact-disc-mech Closed-Loop Balanced Reduction with Application to a Compact Disc Mechanism]. Selected Topics in Identification, Modeling and Control, 9: 47--58, 1996.</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>

Butterfly Gyroscope

2023-11-30T09:26:47Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:ODE]]
[[Category:Linear]]
[[Category:Time invariant]]
[[Category:Second differential order]]
[[Category:MIMO]]
[[Category:Sparse]]

{{Infobox
|Title = Butterfly Gyroscope
|Benchmark ID = butterflyGyroscope_n17361m1q12
|Category = oberwolfach
|System-Class = LTI-SOS
|nstates = 17361
|ninputs = 1
|noutputs = 12
|nparameters = 0
|components = B, C, E, K, M
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
* [[User:Mlinaric]]
* [[User:Hund]]
|Zenodo-link = NA
}}

==Description==
<figure id="fig1">[[File:Butterfly1.jpg|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Butterfly2.jpg|490px|thumb|right|Figure 2]]</figure>
<figure id="fig3">[[File:Butterfly3.jpg|490px|thumb|right|Figure 3]]</figure>

The '''butterfly gyro''' is developed at the [http://www.imego.com Imego Institute] in an ongoing project with [[wikipedia:Saab_Bofors_Dynamics|Saab Bofors Dynamics AB]].
The butterfly is a vibrating micro-mechanical gyro that has sufficient theoretical performance characteristics to make it a promising candidate for use in inertial navigation applications.
The goal of the current project is to develop a micro unit for inertial navigation that can be commercialized in the high-end segment of the rate sensor market.
This project has reached the final stage of a three-year phase where the development and research efforts have ranged from model-based signal processing, via electronics packaging to design and prototype manufacturing of the sensor element.
The project has also included the manufacturing of an [[wikipedia:Application-specific_integrated_circuit|ASIC]], named µSIC, that has been especially designed for the sensor (see Fig. 1).

The gyro chip consists of a three-layer silicon wafer stack, in which the middle layer contains the sensor element.
The sensor consists of two wing pairs that are connected to a common frame by a set of beam elements (see Fig. 2 and Fig. 3);
this is the reason the gyro is called the butterfly.
Since the structure is manufactured using an anisotropic wet-etch process, the connecting beams are slanted.
This makes it possible to keep all electrodes, both for capacitive excitation and detection, confined to one layer beneath the two wing pairs.
The excitation electrodes are the smaller dashed areas shown in Fig. 2.
The detection electrodes correspond to the four larger ones.
By applying DC-biased AC voltages to the four pairs of small electrodes, the wings are forced to vibrate in anti-phase in the wafer plane.
This is the excitation mode.
As the structure rotates about the axis of sensitivity (see Fig. 2), each of the masses will be affected by a Coriolis acceleration.
This acceleration can be represented as an inertial force that is applied at right angles with the external angular velocity and the direction of motion of the mass. The Coriolis force induces an anti-phase motion of the wings out of the wafer plane.
This is the detection mode. The external angular velocity can be related to the amplitude of the detection mode, which is measured via the large electrodes.

When planning for and making decisions on future improvements of the butterfly, it is important to improve the efficiency of the gyro simulations.
Repeated analyses of the sensor structure have to be conducted with respect to a number of important issues.
Examples of such are sensitivity to shock, linear and angular vibration sensitivity, reaction to large rates and/or acceleration, different types of excitation load cases, and the effect of force-feedback.

The use of model order reduction indeed decreases runtimes for repeated simulations.
Moreover, the reduction technique enables a transformation of the FE representation of the gyro into a state space equivalent formulation.
This will prove helpful in testing the model-based Kalman signal processing algorithms that are being designed for the '''butterfly gyro'''.

The structural model of the gyroscope has been done in [http://www.ansys.com/ ANSYS] using quadratic tetrahedral elements (SOLID187, see Fig. 3).
The model shown is a simplified one with a coarse mesh as it is designed to test the model reduction approaches.
It includes the pure structural mechanics problem only. The load vector is composed of time-varying nodal forces applied at the centers of the excitation electrodes (see Fig. 2).
The amplitude and frequency of each force are equal to <math>0.055 \, \mu \text{N}</math> and <math>2384 \, \text{Hz}</math>, respectively.
The Dirichlet boundary conditions have been applied to all degrees of freedom of the nodes belonging to the top and bottom surfaces of the frame.
The output nodes are listed in Table 2 and correspond to the centers of the detection electrodes (see Fig. 3).
The structural model

:<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>

contains the mass <math>M</math> and stiffness matrices <math>K</math>.
The damping matrix <math>E</math> can be modeled as <math>E = \alpha M + \beta K</math>, where the typical values of <math>\alpha</math> and <math>\beta</math> are <math>0</math> and <math>10^{-6}</math> respectively.
The nature of the damping matrix is in reality more complex (squeeze film damping, thermo-elastic damping, etc.) but this simple approach has been chosen with respect to the model reduction test.
<math>B</math> is the load vector, <math>C</math> is the output matrix.

The statistics for the matrices are shown in Table 1.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: System matrices for the gyroscope.''
|-
!Matrix
!m
!n
!nnz
!Is Symmetric?
|-
|M
|17361
|17361
|178896
|Yes
|-
|K
|17361
|17361
|519260
|Yes
|-
|B
|17361
|1
|8
|No
|-
|C
|12
|17361
|12
|No
|}

The outputs are detailed in Table 2.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 2. Outputs for the Butterfly Gyro Model.''
|-
!Index
!Code
!Comment
|-
|1-3
|det1m_Ux, det1m_Uy, det1m_Uz
|Displacements of detection electrode 1, (bottom left large electrode of Fig. 2)
|-
|4-6
|det1p_Ux, det1p_Uy, det1p_Uz
|Displacements of detection electrode 2, (bottom right large electrode of Fig. 2)
|-
|7-9
|det2m_Ux, det2m_Uy, det2m_Uz
|Displacements of detection electrode 3, (top left large electrode of Fig. 2)
|-
|10-12
|det2p_Ux, det2p_Uy, det2p_Uz
|Displacements of detection electrode 4, (top right large electrode of Fig. 2)
|}

The model reduction of the gyroscope model by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem] is described in
<ref name="lienemann04"/>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 35889, see: <ref name="billger05"/>.

==Data==

Download matrices in [http://math.nist.gov/MatrixMarket/ Matrix Market] format:
* [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ButterflyGyro-dim1e5-gyro.tar.gz ButterflyGyro-dim1e5-gyro.tar.gz] (7.4 MB)

The matrix name is used as an extension of the matrix file. File <tt>*.C.names</tt> contains a list of output names written consecutively.

==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}^{17361 \times 17361}</math>,
<math>E \in \mathbb{R}^{17361 \times 17361}</math>,
<math>K \in \mathbb{R}^{17361 \times 17361}</math>,
<math>B \in \mathbb{R}^{17361 \times 1}</math>,
<math>C \in \mathbb{R}^{12 \times 17361}</math>.

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

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection, '''Butterfly Gyroscope'''. hosted at MORwiki - Model Order Reduction Wiki, 2004. http://modelreduction.org/index.php/Butterfly_Gyroscope

@MISC{morwiki_gyro,
author = <nowiki>{{Oberwolfach Benchmark Collection}}</nowiki>,
title = {Butterfly Gyroscope},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Butterfly_Gyroscope}</nowiki>,
year = 2004
}

* For the background on the benchmark:

@INPROCEEDINGS{morBil05,
author = {D. Billger},
title = {The Butterfly Gyro},
booktitle = {Dimension Reduction of Large-Scale Systems},
publisher = {Springer-Verlag, Berlin/Heidelberg, Germany},
year = 2005,
volume = 45,
pages = {349--352},
series = {Lecture Notes in Computational Science and Engineering},
doi = {10.1007/3-540-27909-1_18}
}

==References==

<references>

<ref name="lienemann04"> J. Lienemann, D. Billger, E.B. Rudnyi, A. Greiner, and J.G. Korvink, [http://modelreduction.com/doc/papers/lienemann04MSM.pdf MEMS Compact Modeling Meets Model Order Reduction: Examples of the Application of Arnoldi Methods to Microsystem Devices], Technical Proceedings of the 2004 Nanotechnology Conference and Trade Show, Nanotech 2004, March 7-11, 2004, Boston, Massachusetts, USA.</ref>

<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="billger05"> D. Billger, [https://doi.org/10.1007/3-540-27909-1_18 The Butterfly Gyro], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 349--352, 2005.</ref>

</references>

Building Model

2023-11-30T09:26:18Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:second differential order]]
[[Category:SISO]]

{{Infobox
|Title = Building Model
|Benchmark ID = buildingModel_n48m1q1
|Category = slicot
|System-Class = LTI-FOS
|nstates = 48
|ninputs = 1
|noutputs = 1
|nparameters = 0
|components = A, B, C
|License = NA
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
* [[User:Mlinaric]]
|Zenodo-link = NA
}}

==Description: Motion Problem in a Building==

This benchmark models the displacement of a multi-story building for example during an Earthquake.
More details can be found in <ref name="antoulas01"/> and <ref name="chahlaoui02"/>, <ref name="chahlaoui05"/>.

===Earthquake Model===

==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> are available from the [https://www.slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [https://www.slicot.org/objects/software/shared/bench-data/build.zip build.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

Here is [https://www.python.org Python] code for loading the matrices (<math>A</math> is stored as a sparse matrix that is mostly full and <math>C</math> is stored as an array of 8-bit unsigned integers):

:<syntaxhighlight lang="python">
import numpy as np
from scipy.io import loadmat

mat = loadmat('build.mat')
A = mat['A'].toarray()
B = mat['B']
C = mat['C'].astype(np.float64)
</syntaxhighlight>

The <math>(A, B, C)</math> represents a second-order system

:<math>
\begin{align}
\ddot{q}(t) + E_{so} \dot{q}(t) + K_{so} q(t) &= B_{so} u(t), \\
y(t) &= C_{so} \dot{q}(t),
\end{align}
</math>

as

:<math>
\begin{align}
A &=
\begin{pmatrix}
0 & I \\
-K_{so} & -E_{so}
\end{pmatrix}, \\
B &=
\begin{pmatrix}
0 \\
B_{so}
\end{pmatrix}, \\
C &=
\begin{pmatrix}
0 & C_{so}
\end{pmatrix}
\end{align}
</math>

Here is [https://www.python.org Python] code for checking the structure and extracting the second-order matrices:

:<syntaxhighlight lang="python">
n = 48
n2 = n // 2

assert np.all(A[:n2, :n2] == 0)
assert np.all(A[:n2, n2:] == np.eye(n2))
assert np.all(B[:n2] == 0)
assert np.all(C[:, :n2] == 0)

Eso = -A[n2:, n2:]
Kso = -A[n2:, :n2]
Bso = B[n2:]
Cso = C[:, n2:]
</syntaxhighlight>

==Dimensions==

===First differential order===

System structure:

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

System dimensions:

<math>A \in \mathbb{R}^{48 \times 48}</math>,
<math>B \in \mathbb{R}^{48 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 48}</math>.

===Second differential order===

System structure:

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

System dimensions:

<math>E, K \in \mathbb{R}^{24 \times 24}</math>,
<math>B \in \mathbb{R}^{24 \times 1}</math>,
<math>C_v \in \mathbb{R}^{1 \times 24}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MANUAL{slicot_build,
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 background on the benchmark:

@ARTICLE{morAntSG01,
author = <nowiki>{A.C. Antoulas, D.C. Sorensen and S. Gugercin}</nowiki>,
title = {A survey of model reduction methods for large-scale systems},
journal = {Contemporary Mathematics},
volume = {280},
pages = {193--219},
year = {2001},
doi = {10.1090/conm/280}
}

==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="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>

Bone Model

2023-11-30T09:25:40Z

Malhotra: infobox string

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:second differential order]]

{{Infobox
|Title = Bone Model
|Benchmark ID =
* boneModelB010
* boneModelB025
* boneModelB050
* boneModelB120
* boneModelBS01
* boneModelBS10
|Category = oberwolfach
|System-Class = LTI-SOS
|nstates =
* 986703
* 2159661
* 4934244
* 11969688
* 127224
* 914898
|ninputs = 1
|noutputs = 3
|nparameters = 0
|components = B, C, K, M
|License = GPLv2
|Creator = [[User:Himpe]]
|Editor =
* [[User:Himpe]]
* [[User:Lund]]
|Zenodo-link = NA
}}

==Description: Trabecular Bone Micro-Finite Element Models==

<figure id="fig1">[[File:Bone1.jpg|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Bone2.jpg|490px|thumb|right|Figure 2]]</figure>

Three-dimensional serial reconstruction techniques allow us to develop a very detailed micro-finite element (micro-FE) model of bones that can very accurately represent the porous bone micro-architecture.
Fig. 1 sketches the micro finite element analysis <ref name="rietbergen1995"/>.
Micro [[wikipedia:CT_scan|computed tomography]] (CT) is employed to make 3D high-resolution images (~50 microns) of a bone.
Then the 3D reconstruction is directly transformed into an equally shaped micro finite element model by simply converting all bone [[wikipedia:Voxel|voxels]] to equally sized 8-node brick elements.
This results in [[wikipedia:Finite_Element_Method|finite element]] (FE) models with a very large number of elements.
Such models can be used, for example, to study differences in bone tissue loading between healthy and [[wikipedia:Osteoporosis|osteoporotic]] human bones during quasi-static loading <ref name="rietbergen2003"/>.

There is increasing evidence, however, that bone responds in particular to dynamic loads <ref name="lanyon1984"/>.
It has been shown that the application of high-frequency, very low-magnitude strains to a bone can prevent bone loss due to osteoporosis and can even result in increased bone strength in bones that are already osteoporotic.
In order to better understand this phenomenon, it is necessary to determine the strain as sensed by the bone cells due to this loading.
This would be possible with the micro-FE analysis, but such an analysis needs to be a dynamic one.

The present benchmark presents six bone models varying in dimension from about two hundred thousand to twelve million equations with the goal to research on scalability of model reduction software.
Each model represents a second-order system of the form

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

where the matrices <math>M</math> and <math>K</math> are symmetric and positive definite.
The goal of model reduction is to speed up harmonic response analysis in the frequency range <math>1-100 Hz</math>.

The matrix properties are given in Table 1 below.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Bone micro-finite element models.''
|-
!
!BS01
!BS10
!B010
!B025
!B050
!B120
|-
|Number of Elements
|20098
|192539
|278259
|606253
|1378782
|3387547
|-
|Number of Nodes
|42508
|305066
|329001
|719987
|1644848
|3989996
|-
|Number of DoFs
|127224
|914898
|986703
|2159661
|4934244
|11969688
|-
|nnz in half M
|1182804
|9702186
|12437739
|27150810
|61866069
|151251738
|-
|nnz in half K
|3421188
|28191660
|36326514
|79292769
|180663963
|441785526
|-
|File
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e6-BS01.dat.gz BoneModel-dim1e6-BS01.dat.gz] (305.1 kB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-BS10.dat.gz BoneModel-dim1e7-BS10.dat.gz] (2.8 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B010.dat.gz BoneModel-dim1e7-B010.dat.gz] (3.9 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B025.dat.gz BoneModel-dim1e7-B025.dat.gz] (8.6 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B050.dat.gz BoneModel-dim1e7-B050.dat.gz] (19.6 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e8-B120.dat.gz BoneModel-dim1e8-B120.dat.gz] (48.5 MB)
|}

It should be stressed that the first two models have been obtained differently and they are much simpler to deal with than the last four.
The connectivity in the last four models is about four times higher.
This can be seen by comparing models <tt>BS10</tt> and <tt>B010</tt>.
Although models look similar by number of nonzeros in the system matrices, the model <tt>B010</tt> is much harder to solve:
The number of nonzero elements in the factor for model <tt>B010</tt> is about four times more than for <tt>BS10</tt>.

The method allows for the compact representation of the models, as the element mass and stiffness matrices are the same for all elements.
As a result, a file describing the node indices for each element is enough to assemble the global matrix.
Each node has three degrees of freedom (<math>UX</math>, <math>UY</math>, <math>UZ</math>) and it contributes three consecutive entries to the state vector.
The node numbering is natural from the first to the last.
The assembly procedure as a pseudo-code is presented below (indices start from one).
It is assumed that the last <math>300</math> degrees of freedom are fixed as zero Dirichlet boundary conditions.
For simplicity, the pseudo-code does not take into account that the matrix is symmetric.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>.

==Data==

The data file for each model contains the number of elements, <tt>nel</tt>, and the number of nodes, <tt>nnod</tt>, in the first line and then <tt>nel</tt> number of lines with eight numbers for node indices in each line.

1) Read the element stiffness matrix elemK(24,24), 8 nodes * 3 degrees of freedom per node.
2) Read the number of elements, nel, and number of node, nnod, from the first line of the data file.
3) Number of degrees of freedom, ndof = nnod *3 - 300.
4) Allocate space for the sparse global matrix, matK(ndof, ndof).
5) Assembly:
Do (k = 1, nel)
Read eight node numbers from the k-th line, nodeindex(8);
Construct the index for degrees of freedom, dofindex(24):
Do (i = 1, 8)
Do (j = 1, 3)
dofindex((i - 1)*3 + j)= (nodeindex(i) - 1)*3 + j;
Use dofindex to assemble the element matrix elemK:
Do (i = 1, 24)
Do (j = 1, 24)
If (dofindex(i) < ndof AND dofindex(j) < ndof)
matK(dofindex(i), dofindex(j)) += elemK(i, j)

The input matrix contains a single column with <math>B(1) = 1</math>.
The output matrix takes the first three components of the state vector, that is, three displacements <math>UX</math>, <math>UY</math>, and <math>UZ</math> for the first node.

The archive [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-assemble.tar.gz BoneModel-assemble.tar.gz] contains the element mass and stiffness matrices as well as the sample code in [[wikipedia:C++|C++]] to assemble the dynamic system.
The code can write the dynamic system in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format or can be used as a hook to transform the global matrices to an appropriate format.
The gzipped compressed data files for element assembly as described above can be downloaded from Table 1.

Model reduction for models <tt>BS010</tt> and <tt>BS10</tt> was performed in <ref name="rudnyi2005"/>. The benchmarking of the parallel [http://graal.ens-lyon.fr/MUMPS/ MUMPS] direct solver <ref name="amestoy2006"/> for the stiffness matrices is described in <ref name="rudnyi2006"/>.

Files (in .mat format) can be found for the three smaller systems ([https://sparse.tamu.edu/Oberwolfach/boneS01 BS01], [https://sparse.tamu.edu/Oberwolfach/boneS10 BS10], and [https://sparse.tamu.edu/Oberwolfach/bone010 B010]) on SuiteSparse.

==Dimensions==

System structure:

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

System dimensions:

<math>M, K \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times 1}</math>,
<math>C \in \mathbb{R}^{3 \times n}</math>.

System variants:

<tt>BS01</tt>: <math>n = 127224</math>,
<tt>BS10</tt>: <math>n = 914898</math>,
<tt>B010</tt>: <math>n = 986703</math>,
<tt>B025</tt>: <math>n = 2159661</math>,
<tt>B050</tt>: <math>n = 4934244</math>,
<tt>B120</tt>: <math>n = 11969688</math>.

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

@ARTICLE{RieWHetal95,
author = <nowiki>{B. van Rietbergen and H. Weinans and R. Huiskes and A. Odgaard}</nowiki>,
title = {A new method to determine trabecular bone elastic properties and loading using micromechanical finite-elements models},
journal = {Journal of Biomechanics},
volume = {28},
number = {1},
pages = {69--81},
year = {1995},
doi = {10.1016/0021-9290(95)80008-5}
}

==References==

<references>

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

<ref name="rietbergen1995">B. van Rietbergen, H. Weinans, R. Huiskes, A. Odgaard, [https://doi.org/10.1016/0021-9290(95)80008-5 A new method to determine trabecular bone elastic properties and loading using micromechanical finite-elements models] Journal of Biomechanics, 28(1): 69--81, 1995.</ref>

<ref name="rietbergen2003">B. van Rietbergen, R. Huiskes, F. Eckstein, P. Rueegsegger, [https://doi.org/10.1359/jbmr.2003.18.10.1781 Trabecular Bone Tissue Strains in the Healthy and Osteoporotic Human Femur], Journal of Bone and Mineral Research, 18(10): 1781--1787, 2003.</ref>

<ref name="lanyon1984">L.E. Lanyon, C.T. Rubin, [https://doi.org/10.1016/0021-9290(84)90003-4 Static versus dynamic loads as an influence on bone remodelling], Journal of Biomechanics, 17: 897--906, 1984.</ref>

<ref name="rudnyi2005">E.B. Rudnyi, B. van Rietbergen, J.G. Korvink. [http://repository.tue.nl/b3454bd9-f190-4c33-8dca-22c50b08edec Efficient Harmonic Simulation of a Trabecular Bone Finite Element Model by means of Model Reduction]. 12th Workshop "The Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields", Proceedings of the 12th FEM Workshop: 61--68, 2005</ref>

<ref name="amestoy2006">P.R. Amestoy, A. Guermouche and J.-Y. L'Excellent, S. Pralet, [https://doi.org/10.1016/j.parco.2005.07.004 Hybrid scheduling for the parallel solution of linear systems]. Parallel Computing, 32(2): 136--156, 2006.</ref>

<ref name="rudnyi2006">E.B. Rudnyi, B. van Rietbergen, J. G. Korvink. [http://modelreduction.com/doc/papers/rudnyi06tam.pdf Model Reduction for High Dimensional Micro-FE Models]. TAM'06, The Third HPC-Europa Transnational Access Meeting, Barcelona, 2006.</ref>

</references>

Artificial Fishtail

2023-11-30T09:24:57Z

Malhotra:

[[Category:benchmark]]
[[Category:ODE]]
[[Category:Linear]]
[[Category:Time invariant]]
[[Category:Second differential order]]
[[Category:Sparse]]
[[Category:MIMO]]

{{Infobox
|Title = Artificial Fishtail
|Benchmark ID =
* artificialFishtail_n779232m1q21
* artificialFishtail_n779232m1q3
|Category = misc
|System-Class = LTI-SOS
|nstates = 779232
|ninputs = 1
|noutputs =
* 21
* 3
|nparameters = 0
|components = B, C, E, K, M
|License = Creative Commons Attribution 4.0 International
|Creator = [[User:Saak]]
|Editor =
* [[User:Saak]]
|Zenodo-link = https://zenodo.org/record/2558728/files/fish_tail.mat
}}

<figure id="fig:plot1">
[[File:Fishtail.png|4380px|thumb|right|<caption>Schematic 3D-Model-Fishtail</caption>]]
</figure>
<figure id="fig:plot2">
[[File:fishtail_tf.png|438px|thumb|right|<caption>Sigma Plot Fishtail Cp</caption>]]
</figure>
<figure id="fig:plot3">
[[File:fishtail_ext_tf.png|438px|thumb|right|<caption>Sigma Plot Fishtail Cp_ext</caption>]]
</figure>

==Description==
Today's [[wikipedia:Autonomous_underwater_vehicle|autonomous underwater vehicles]] (AUVs) are a source of noise pollution and inefficiency due to their screw propeller-driven design.
The evolution of fish has, on the other hand, optimized their underwater efficiency and agility over millennia.
The adaption of fish-like drive systems for AUVs is therefore a promising choice.

==Model Description==
This model describes the silicon body of an artificial fishtail supported by a central carbon beam.
The rear part of the fish body without the fins is modeled as a 3D FEM model using linear elasticity.
In the current stage of modeling the tail is rigidly mounted in the front, the states in <math>x</math> represent the displacements of the finite element degrees of freedom.
The fish-like locomotion is enabled by pumping air between two sets of pressure chambers in the left and right halves of the tail.
The single input <math>u</math> of the system is thus the pumping pressure.
The outputs are displacements of certain surface points.
There are two variants of the model.
The first has three outputs representing the displacements of the point of interest, the rear tip of the carbon beam, in the three spatial directions.
For the second variant, six additional points <math>(z_1,z_2,z_3)</math> on the flank are added as outputs, yielding a total of 21 outputs.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
! scope="col" style="width: 12ex;" | <math>z_1</math>
! scope="col" style="width: 12ex;" | <math>z_2</math>
! scope="col" style="width: 12ex;" | <math>z_3</math>
|-
| 0.05
| 0.0
| 0.0
|-
| 0.0474526
| 0.0
| 0.0599584
|-
| 0.04032111
| 0.0
| 0.105274
|-
| 0.0326229
| 0.0
| 0.136726
|-
| 0.0250675
| 0.0
| 0.16107
|-
| 0.0168069
| 0.0
| 0.183588
|-
| 0.0
| 0.0
| 0.21
|}

Note that the POI (Point of Interest) is the last row in this table and in Cp_ext in the data files (see below).
The additional outputs show two effects.
On the one hand, for purely input-output-related reduction methods they avoid drastic deviations on the interior states.
On the other hand, they show a smoothing effect for the model's transfer function.

==Origin==
The model was set up and computed at the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Artificial_Fishtail chair of automatic control] at CAU Kiel and first presented in <ref name="SieKM18" />.

==Data==
The model is based on the finite element package [https://www.firedrakeproject.org Firedrake] and uses the material parameters:
{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Part
|Parameter
|Value
|Unit
|-style="background-color:#FFFFFF;"
|
|<math>\varrho_1</math>
|<math>1.07 \cdot 10^{-3}</math>
|<math>\frac{\text{kg}}{\text{m}^{3}}</math>
|-style="background-color:#FFFFFF;"
|Hull
|<math>E_1</math>
|<math>0.025 \cdot 10^6</math>
|<math>\frac{\text{kg}}{\text{m}\text{s}^2}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\nu_1</math>
|<math>0.48</math>
|
|-
|
|<math>\varrho_2</math>
|<math>1.4 \cdot 10^{3}</math>
|<math>\frac{\text{kg}}{\text{m}^{3}}</math>
|-
| Beam
|<math> E_2</math>
|<math> 2.96 \cdot 10^{10}</math>
|<math> \frac{\text{kg}}{\text{m}\text{s}^2}</math>
|-
|
|<math>\nu_2</math>
|<math> 0.3</math>
|
|-style="background-color:#FFFFFF;"
| Rayleigh damping
|<math>\alpha_r</math>
|<math> 1.0 \cdot 10^{-4}</math>
|<math>\frac{1}{s}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\beta_r</math>
|<math> 2.0 \cdot 10^{-4}</math>
|<math>\text{s}</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, E, K \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 = 779\,232</math> and <math>m = 1</math>.

The internal damping is modeled as Rayleigh damping <math>E = \alpha_r M + \beta_r K</math> using the coefficients from the table above.

System variants:

* <math>p = 3</math>: <math>C=</math><tt>Cp</tt> in the [https://doi.org/10.5281/zenodo.2558728 data files],
* <math>p = 21</math>: <math>C=</math><tt>Cp_ext</tt> in the [https://doi.org/10.5281/zenodo.2558728 data files].

== Remarks ==
* Physically meaningful inputs are of dimension <math>u(t) = \mathcal{O}(10^3)</math>. As an example, a step signal with around <math>5\,000</math>Pa leads to a horizontal POI displacement of about <math>3</math>cm.
* The interesting operation frequencies are in the range between <math>0</math>Hz and <math>10</math>Hz.
* If required, the finite element mesh behind the model and a CSV file with the output locations are available [https://doi.org/10.5281/zenodo.2565173 separately].
* Warning: the data files are quite large and may exceed the RAM of a typical machine if the user is also running MATLAB.

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

* For the benchmark itself and its data:
@Misc{SieKM19,
author = {Siebelts, D. and Kater, A. and Meurer, T. and Andrej, J.},
title = {Matrices for an Artificial Fishtail},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2019,
doi = {10.5281/zenodo.2558728}
}

* For the background on the benchmark:

@Article{SieKM18,
author = {Siebelts, D. and Kater, A. and Meurer, T.},
title = {Modeling and Motion Planning for an Artificial Fishtail},
journal = {IFAC-PapersOnLine},
volume = 51,
number = 2,
year = 2018,
pages = {319--324},
doi = {10.1016/j.ifacol.2018.03.055},
publisher = {Elsevier {BV}}
}

==References==
<references>
<ref name="SieKM18">D. Siebelts, A. Kater, T. Meurer, [https://doi.org/doi:10.1016/j.ifacol.2018.03.055 Modeling and Motion Planning for an Artificial Fishtail], IFAC-PapersOnLine (9th Vienna International Conference on Mathematical Modelling) 51(2): 319--324, 2018.</ref>
</references>

Anemometer

2023-11-30T09:24:16Z

Malhotra:

[[Category:benchmark]]
[[Category:parametric]]
[[Category:linear]]
[[Category:ODE]]
[[Category:time invariant]]
[[Category:Parametric]]
[[Category:affine parameter representation]]
[[Category:first differential order]]
[[Category:SISO]]

{{Infobox
|Title = Anemometer
|Benchmark ID =
* anemometer1Param_n29008m1q1
* anemometer3Param_n29008m1q1
|Category = misc
|System-Class = AP-LTI-FOS
|nstates = 29008
|ninputs = 1
|noutputs = 1
|nparameters =
* 2
* 5
|components = A, B, C, E
|License = NA
|Creator = [[User:Feng]]
|Editor =
* [[User:Feng]]
* [[User:Himpe]]
* [[User:Lnor]]
* [[User:Baur]]
* [[User:Will]]
* [[User:Lund]]
|Zenodo-link = NA
}}

==Description==

<figure id="fig:plot1">
[[File:Model_Color.pdf|600px|thumb|right|<caption>Schematic 2D-Model-Anemometer</caption>]]
</figure>
<figure id="fig:plot2">

[[file:ContourPlot30.pdf|600px|thumb|right|<caption>Calculated temperature profile for the Anemometer function</caption>]]

</figure>

An '''Anemometer'''<ref name="ernst01" group="a)"/><ref name="benner05" group="a)"/><ref name="moosmann05" group="a)"/><ref name="moosmann07" group="c)"/><ref name="moosmann05b" group="c)"/><ref name="rudnyi06" group="c)"/> (see [[wikipedia:Thermal_mass_flow_meter|thermal mass flow meter]])
is a flow sensing device, consisting of a heater and temperature sensors before and after the heater, placed either directly in the flow or in its vicinity Fig. 1.
They are located on a membrane to minimize heat dissipation through the structure.
Without any flow, the heat dissipates symmetrically into the fluid.
This symmetry is disturbed if a flow is applied to the fluid,
which leads to a convection on the temperature field and therefore to a difference between the temperature sensors (see Fig. 2) from which the fluid velocity can be determined.

The physical model can be expressed by the [[wikipedia:Convection–diffusion_equation|convection-diffusion partial differential equation]] <ref name="moosmann04" group="b)"/>:

:<math>
\rho c \frac{\partial T}{\partial t}
= \nabla \cdot (\kappa \nabla T)
- \rho c v \nabla T
+ \dot{q},
</math>

where <math>\rho</math> denotes the mass density, <math>c</math> is the specific heat capacity, <math>\kappa</math> is the thermal conductivity,
<math>v</math> is the fluid velocity, <math>T</math> is the temperature, and <math>\dot q</math> is the heat flow into the system caused by the heater.

The solid model has been generated and meshed in [[wikipedia:ANSYS|ANSYS]].
Triangular [http://www.ansys.stuba.sk/html/elem_55/chapter4/ES4-55.htm PLANE55] elements have been used for the finite element discretization.
The order of the system is <math>n = 29008</math>.

Example with one parameter:

The <math>n</math> dimensional [[wikipedia:Ordinary_Differential_Equation|ODE]] system has the following transfer function
:<math>
G(s, p) = C (s E - A_1 - p (A_2 - A_1))^{-1} B
</math>

with the fluid velocity <math>p(=v)</math> as single parameter.
Here <math>E</math> is the heat capacitance matrix, <math>B</math> is the load vector which is derived from separating the spatial and temporal variables in <math>\dot{q}</math> and the [[wikipedia:Finite_Element_Method|FEM]] discretization w.r.t. the spatial variables.
<math>A_i</math> are the stiffness matrices with <math>i=1</math> for pure diffusion and <math>i=2</math> for diffusion and convection.
Thus, for obtaining pure convection you have to compute <math>A_2 - A_1</math>.

Example with three parameters:

Here, all fluid properties are identified as parameters. Thus, we consider the following transfer function

:<math>
G(s, p_0, p_1, p_2) =
C
(
s \underbrace{(E_s + p_0 E_f)}_{E(p_0)}
- \underbrace{(A_{d,s} + p_1 A_{d,f} + p_2 A_c)}_{A(p_1,p_2)}
)^{-1}
B
</math>

with parameters <math>p_0</math>, <math>p_1</math>, <math>p_2</math> which are combinations of the original fluid parameters <math>\rho</math>, <math>c</math>, <math>\kappa</math>, <math>v</math>: <math>p_0 = \rho c</math>, <math>p_1 = \kappa</math>, and <math>p_2 = \rho c v</math>, see <ref name="baur11" group="c)"/>. So far, we have considered the mass density as fixed, i.e. <math>\rho=1</math>.

==Origin==

* [http://www.imtek.uni-freiburg.de/professuren/simulation/ IMTEK Freiburg, Simulation group], Prof Dr Jan G. Korvink has taken on a position as Director of the Institute of Microstructure Technology (IMT) at the Karlsruhe Institute of Technology (KIT).

==Data==

Matrices are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format.
All matrices (for the one parameter system and for the three parameter case) can be found and uploaded in [[Media:Anemometer.tar.gz|Anemometer.tar.gz]].
The matrix name is used as an extension of the matrix file.
The system matrices have been extracted from ANSYS models by means of [http://simulation.uni-freiburg.de/downloads/mor4fem mor4fem].
For more information about computing the system matrices, the choice of the output, applying the permutation, please look into the [[media:Readme2.pdf|readme file]]. [[File: Readme2.pdf|thumb]]

Example with one parameter:

* <tt>.B</tt>: load vector
* <tt>.E</tt>: heat capacitance matrix
* <tt>.P</tt>: permutation matrix
* <tt>.A</tt>: stiffness matrices (2)

Example with three parameters:

* <tt>.B</tt>: load vector
* <tt>.E</tt>: heat capacitance matrices (2)
* <tt>.A</tt>: stiffness matrices (5)

To test the quality of the reduced order systems, harmonic simulations as well as transient step responses could be computed, see <ref name="baur11" group="c)"/>.

The output matrix <math>C \in \mathbb{R}^{1 \times 29008}</math> is a vector with non-zero elements <math>C_{173} = 1</math> and <math>C_{133} = -1</math>.

==Dimensions==

System structure (1 parameter):

:<math>
\begin{align}
E \dot{x}(t) &= (A_1 + p (A_2 - A_1)) x(t) + B u(t) \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>E \in \mathbb{R}^{29008 \times 29008}</math>,
<math>A_{1,2} \in \mathbb{R}^{29008 \times 29008}</math>,
<math>B \in \mathbb{R}^{29008 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 29008}</math>.

System structure (3 parameter):

:<math>
\begin{align}
(E_1 + p_0 (E_2 - E_1)) \dot{x}(t) &= (A_1 + p_1 (A_3 - A_1 + A_4 - A_5) + p_2 (A_2 - A_1)) x(t) + B u(t) \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>E_{1,2} \in \mathbb{R}^{29008 \times 29008}</math>,
<math>A_{1,2,3,4,5} \in \mathbb{R}^{29008 \times 29008}</math>,
<math>B \in \mathbb{R}^{29008 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 29008}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_anemom,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Anemometer},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Anemometer}</nowiki>,
year = {2018}
}

* For the background on the benchmark:

==References==

a) About the '''Anemometer'''

<references group="a)">

<ref name="ernst01" group="a)">H. Ernst, "[http://www.freidok.uni-freiburg.de/volltexte/201/ High-Resolution Thermal Measurements in Fluids]," PhD thesis, University of Freiburg, Germany (2001).</ref>

<ref name="benner05" group="a)">P. Benner, V. Mehrmann and D. Sorensen, "[http://dx.doi.org/10.1007/3-540-27909-1 Dimension Reduction of Large-Scale Systems]", Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin/Heidelberg, Germany, 45, 2005.</ref>

<ref name="moosmann05" group="a)">C. Moosmann and A. Greiner, "[http://dx.doi.org/10.1007/3-540-27909-1_16 Convective Thermal Flow Problems]", Chapter 16 (pages 341--343) of 2.</ref>

</references>

b) MOR for non-parametrized '''Anemometer'''

<references group="b)">

<ref name="moosmann04" group="b)">C. Moosmann, E. B. Rudnyi, A. Greiner and J. G. Korvink, "[http://modelreduction.com/doc/papers/moosmann04THERMINIC.pdf Model Order Reduction for Linear Convective Thermal Flow]",
Proceedings of 10th International Workshops on THERMal INvestigations of ICs and Systems, THERMINIC2004, 29 Sept - 1 Oct, 2004, Sophia Antipolis, France.</ref>

</references>

c) MOR for parametrized '''Anemometer'''

<references group="c)">

<ref name="baur11" group="c)">U. Baur, P. Benner, A. Greiner, J. G. Korvink, J. Lienemann and C. Moosmann, "[http://dx.doi.org/10.1080/13873954.2011.547658 Parameter preserving model order reduction for MEMS applications]", MCMDS Mathematical and Computer Modeling of Dynamical Systems, 17(4):297--317, 2011.</ref>

<ref name="moosmann07" group="c)">C. Moosmann, "[http://www.freidok.uni-freiburg.de/volltexte/3971/ ParaMOR - Model Order Reduction for parameterized MEMS applications]", PhD thesis, University of Freiburg, Germany (2007).</ref>

<ref name="moosmann05b" group="c)">C. Moosmann, E. B. Rudnyi, A. Greiner, J. G. Korvink and M. Hornung, "[http://modelreduction.com/doc/papers/moosmann05MSM.pdf Parameter Preserving Model Order Reduction of a Flow Meter]", Technical Proceedings of the 2005 Nanotechnology
Conference and Trade Show, Nanotech 2005, May 8-12, 2005, Anaheim, California, USA, NSTINanotech
2005, vol. 3, p. 684-687.</ref>

<ref name="rudnyi06" group="c)">E. B. Rudnyi, C. Moosmann, A. Greiner, T. Bechtold, J. G. Korvink, "[http://modelreduction.com/doc/papers/rudnyi06mathmod.pdf Parameter Preserving Model Reduction for MEMS System-level Simulation and Design]", Proceedings of MATHMOD 2006, February 8 -
10, 2006, Vienna University of Technology, Austria.</ref>

</references>

==Contact==

[[User:Himpe]]

Power system examples

2023-11-23T11:06:40Z

Malhotra: Add correct infobox

[[Category:benchmark]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:Differential_algebraic_system]]

{{Infobox
|Title = Power System Examples
|Benchmark ID =
* bauru5727_n40366m2q2
* bips071693_n13275m4q4
* bips071998_n15066m4q4
* bips072476_n16861m4q4
* bips073078_n21128m4q4
* bips1997_n13250m1q1
* bips2007_n21476m32q32
* bips97_n13251m1q1
* bips97mimo28_n13251m28q28
* bips97mimo46_n13250m46q46
* bips97mimo8_n13309m8q8
* bips981142_n9735m4q4
* bips981450_n11305m4q4
* bips98606_n7135m4q4
* juba5723_n40337m2q1
* newEngland_n66m1q1
* nopss_n11685m1q1
* xingo6u_n20738m1q6
* zerosnopss_n13296m46q46
|Category = power_system
|System-Class = LTI-FOS
|nstates =
* 40366
* 13275
* 15066
* 16861
* 21128
* 13250
* 21476
* 13251
* 13251
* 13250
* 13309
* 9735
* 11305
* 7135
* 40337
* 66
* 11685
* 20738
* 13296
|ninputs =
* 2
* 4
* 4
* 4
* 4
* 1
* 32
* 1
* 28
* 46
* 8
* 4
* 4
* 4
* 2
* 1
* 1
* 1
* 46
|noutputs =
* 2
* 4
* 4
* 4
* 4
* 1
* 32
* 1
* 28
* 46
* 8
* 4
* 4
* 4
* 1
* 1
* 1
* 6
* 46
|nparameters = 0
|components =
* A, B, C, E
* A, B, C, E
* A, B, C, E
* A, B, C, E
* A, B, C, E
* A, B, C, E
* A, B, C, D, E
* A, B, C, E
* A, B, C, E
* A, B, C, E
* A, B, C, E
* A, B, C, E
* A, B, C, E
* A, B, C, E
* A, B, C, E
* A, B, C
* A, B, C, D, E
* A, B, C, E
* A, B, C, E
|License = NA
|Creator = [[User:Rommes]]
|Editor =
* [[User:Rommes]]
* [[User:Kuerschner]]
* [[User:Himpe]]
|Zenodo-link = NA
}}

__NUMBEREDHEADINGS__

==Description==

These first order systems are given in generalized state space form

:<math>
E\dot{x}(t)=A x(t)+B u(t), \quad
y(t)=Cx(t)+Du(t),\quad E,A\in\mathbb{R}^{n\times n},~B\in\mathbb{R}^{n\times m},~C\in\mathbb{R}^{p\times n},~D\in\mathbb{R}^{p\times m}\quad(1)
</math>

and originated at [http://www.cepel.br/ CEPEL] for simulating large '''power systems'''.

They come in different sizes and variants, including both SISO and MIMO systems having regular or singular <math>E</math> matrices. In the latter case the DAEs are of index 1 (except for PI Sections 20--80 which are index 2) and using simple row and column permutations, <math>E,A,B,C</math> can be brought into the form

:<math>
E=\left[ \begin{array}{cc}I_{n_f}&0\\0&0\end{array}\right],\quad A=\left[ \begin{array}{cc}A_{11}&A_{12}\\A_{21}&A_{22}\end{array}\right],\quad B=\left[ \begin{array}{cc}B_{1}\\B_2\end{array}\right],\quad C=\left[ \begin{array}{cc}C_{1}&C_2\end{array}\right],
</math>

where <math>n_f</math> denotes the number of finite eigenvalues in <math>\Lambda(A,E)</math> and <math>A_{22}\in\mathbb{R}^{n-n_f\times n-n_f}</math> is regular.
A complete overview over these systems can be found in table below. The '''power systems''' served as benchmark examples for [[Modal truncation|Dominant Pole based Modal Truncation]]<ref name="MarLP96"></ref><ref name="RomM06a"></ref><ref name="RomM06b"></ref><ref name="Rom07"></ref><ref name="RomM08"></ref><ref name="Kue10"></ref> and for a special adaption<ref name="FreRM08"></ref> of [[Balanced Truncation]] for the index-1 DAE systems.

==Data==
The table below lists the charateristics of all '''power systems'''. The files can be downloaded at [https://sites.google.com/site/rommes/software https://sites.google.com/site/rommes/software].

{| class="wikitable sortable" style="text-align: center; width: auto;"
|-
! Name
! <math>n</math>
! <math>m</math>
! <math>p</math>
! Type
|-
|New England
|66
|1
|1
|ODE
|-
|BIPS/97
|13251
|1
|1
|DAE
|-
|BIPS/1997
|13250
|1
|1
|DAE
|-
|BIPS/2007
|21476
|32
|32
|DAE
|-
|BIPS/97,MIMO8
|13309
|8
|8
|DAE
|-
|BIPS/97,MIMO28
|13251
|28
|28
|DAE
|-
|BIPS/97,MIMO46
|13250
|46
|46
|DAE
|-
|Juba5723
|40337
|2
|1
|DAE
|-
|Bauru5727
|40366
|2
|2
|DAE
|-
|zeros_nopss
|13296
|46
|46
|DAE
|-
|xingo6u
|20738
|1
|6
|DAE
|-
|nopss
|11685
|1
|1
|DAE
|-
|bips98_606
|7135
|4
|4
|DAE
|-
|bips98_1142
|9735
|4
|4
|DAE
|-
|bips98_1450
|11305
|4
|4
|DAE
|-
|bips07_1693
|13275
|4
|4
|DAE
|-
|bips07_1998
|15066
|4
|4
|DAE
|-
|bips07_2476
|16861
|4
|4
|DAE
|-
|bips07_3078
|21128
|4
|4
|DAE
|-
|PI Sections:
|
|
|
|
|-
|S10
|682
|1
|1
|DAE
|-
|S20
|1182
|1
|1
|DAE
|-
|S40
|2182
|1
|1
|DAE
|-
|S80
|4182
|1
|1
|DAE
|-
|M10
|682
|3
|3
|DAE
|-
|M20
|1182
|3
|3
|DAE
|-
|M40
|2182
|3
|3
|DAE
|-
|M80
|4182
|3
|3
|DAE
|-
|}

==Background==
Electrical '''power systems''' experience several steady-state and dynamic phenomena that may hinder its reliable, stable operation if not properly designed and operated. Among the dynamic phenomena, '''power system''' angle stability is of major concern, its studies requiring both numerical integration of a stiff set of large, nonlinear differential-algebraic (DAE) equations (for transient stability simulations) and the eigensolution as well as the use of other numerical linear algebra (NLA) algorithms applied to the linearized DAE equations of the same large stability models. '''Power systems''' with multiple electrical power plants, consumer loads, and industrial loads experience complex electromechanical oscillations, much as spring-mass mechanical systems experience mechanical oscillations. These oscillations, when the electric '''power system''' is under stressed conditions, may become poorly damped or unstable. The analysis of these low-frequency oscillations, as well as their controller-induced damping control, is enhanced by the results from NLA algorithms. The study of all these issues constitutes the field known as small signal stability.

There is a pressing need for better utilization of the transmission network and its cost-effective expansions to reliably carry electric power from the generating plants, driven by several primary energy sources, to the loads. This imposes stricter requirements on network design and '''power system''' control equipment, as well as on the adopted control laws, which, coupled with the continental dimensions of modern interconnected '''power systems''', must be studied with the help of simulators employing advanced NLA algorithms. This wiki page contains system models developed when dealing with these NLA issues, with emphasis on the analysis and control of small signal stability.

Modal analysis has been used for Model Order Reduction (MOR) of lightly damped systems like flexible mechanical structures and RLC networks from either '''power systems''' or microprocessor interconnectors. The advent of subspace accelerated dominant pole algorithms in <ref name="MarLP96"></ref><ref name="RomM06a"></ref><ref name="RomM06b"></ref>, made modal reduction effective for better damped systems, such as interconnected '''power systems''', which have eigenvalue clusters in the <math>0.2</math> to <math>2.0 Hz</math> range and damping ratios between <math>-0.05</math> and <math>0.25</math>. Important characteristics of model reduction methods include preserving the dominant dynamics and stability in the Reduced Order Models (ROMs).

Model reduction is important to the linear control system analysis and design of '''power system''' dynamic models. Reduced-order '''power system''' transfer function models are effective in lowering the computational costs of controller design. Once the dominant transfer function poles have been found, the calculation of their associated residues can be obtained by the scalar product of the left and right eigenvectors by the input and output vectors of the transfer function. From the knowledge of the dominant pole-residue set, one can build a transfer function ROM whose accuracy varies with the number of retained poles. See, e.g., the SADPA, SAMDP papers <ref name="RomM06a"></ref><ref name="RomM06b"></ref>.

Modal analysis is also used in the harmonic distortion analysis of '''power system''' networks, mainly in subtransmission and distribution voltage levels. In this application, modal analysis allows for determining the main network parameters that impact distortion levels and suggests changes to their effective values among other alternatives. The linear RLC network is modeled unloaded, the conventional electrical generators can be entirely neglected and the nonlinear loads of various nature are modeled as current injections into the network buses -3rd, 5th, 7th, 11th, 13th harmonics of the nominal system frequency (<math>50 Hz</math> or <math>60 Hz</math>). An even more important application for the modal analysis of these large descriptor system RLC network models is the production of ROM for multiport electrical networks, generally in the form of a state-space system whose dimension is considerably smaller than that of the original system. When these ROMs take the form of RLC networks, they are referred to as RLC reduced equivalents. These ROMs, or equivalents, are used in real-time and off-line simulators for the study of '''power system''' high-frequency transient phenomena. They are produced in varying degrees of complexity by using advanced NLA algorithms, which is an area of intense research work.

==Test systems for small-signal stability analysis of large electric power system networks==
All test '''power systems''' in [https://sites.google.com/site/rommes/software https://sites.google.com/site/rommes/software], except the PI sections example, are used for '''power system''' small-signal stability studies, and several of them are briefly described in <ref name="MarLP96"></ref>. Such studies include stability analysis, controller design, computation of MOR, etc. The dynamic phenomena of interest require the detailed modeling of the electrical energy generators (mainly the large sized ones) and other important devices such as flexible AC transmission systems (FACTS). The 3-phase alternating current (AC) network is comprised of nodes (electrical buses, which may be the representation of an entire electrical energy transmission (or distribution) substation, and branches (transmission lines, transformers, series capacitors) and need have only a static model in this application except for the occasional high-power electronic HVDC (High voltage direct current) transmission links and FACTS devices.

The input <math>u(t)</math> depends on the study of interest, but the excitation system voltage reference is a frequent input variable in small-signal stability studies. Another reference is the active power setpoint of a rotor-speed regulator. The input signal used in all the above mentioned files is of Vref type. The output could be an entry of the generalized state <math>x(t)</math> (state or algebraic variable) or a linear combination of these variables. The generalized states <math>x(t)</math> are associated with equations of generators, controllers and network. The interconnected network has only algebraic variables, except when FACTS devices are connected.
The test systems are represented by generalized state-space models, see <math>(1)</math>, where the feed through matrix <math>D</math> is generally zero.

==Test systems for electromagnetic transients and harmonic distortion studies==
Depending on the harmonic study at hand, a voltage or current source can be assigned as an input. In <ref name="MarLP96"></ref> the current injection <math>u(t)</math> at a node (also, electrical bus, in '''power system''' terminology) is the input, while the nodal voltage, at the same node, is the output (variable <math>y(t)</math>).

All transmission lines in the network are modeled by RLC ladder networks, of cascaded RLC PI-circuits, having fixed parameters <ref name="Wat03"></ref>. The transformers are modeled by series RL circuits. Loads and shunt elements are represented by series-connected RL (or RC) branch, or just an L (or C). As a consequence, the states <math>x(t)</math> of the dynamic system are either the current through an inductor or the voltage across a capacitor. So matrices <math>A, B, C, D, E</math> describe how each circuit element equation, node and voltage are connected, according to the Kirchhoff’s law.

On https://sites.google.com/site/rommes/software the systems labeled by PI Sections 20--80 are of index-2 ([https://drive.google.com/file/d/1ttJgCZeNEspA3Ty7vNPtsuPqo_-Qa-6g/view SISO_PI_n.zip and MIMO_PI_n.zip]). The data of these test systems can be converted into index-0 systems by applying a procedure for symbolic math elimination of algebraic variables and redundant state variables. This procedure is detailed in the Appendix C of <ref name="FreRM11"></ref>.

==References==
<references>
<ref name="MarLP96">N. Martins, L. Lima, and H. Pinto, "[https://doi.org/10.1109/59.486093 Computing dominant poles of power system transfer functions]", IEEE Transactions on
Power Systems, 11: 162--170, 1996.</ref>

<ref name="RomM06a">J. Rommes and N. Martins, "[https://doi.org/10.1109/TPWRS.2006.876671 Efficient computation of transfer function dominant poles using subspace acceleration]", IEEE Transactions on
Power Systems, 21(3): 1218--1226, 2006.</ref>

<ref name="RomM06b">J. Rommes and N. Martins, "[https://doi.org/10.1109/TPWRS.2006.881154 Efficient computation of multivariable transfer function dominant poles using subspace acceleration]", IEEE Transactions on
Power Systems, 21(4): 1471--1483, 2006.</ref>

<ref name="Rom07">J. Rommes, "[https://dspace.library.uu.nl/handle/1874/21787 Methods for eigenvalue problems with applications in model order reduction]", Ph.D. dissertation, Universiteit
Utrecht, 2007.</ref>

<ref name="RomM08">J. Rommes and N. Martins, "[https://doi.org/10.1137/070684562 Computing transfer function dominant poles of large second-order dynamical systems]" SIAM Journal on Scientific Computing, 30(4): 2137--2157, 2008.</ref>

<ref name="FreRM08">F. Freitas, J. Rommes, and N. Martins, "[https://doi.org/10.1109/TPWRS.2008.926693 Gramian-based reduction method applied to large sparse power system descriptor models]" IEEE Transactions on Power Systems, 23(3): 1258--1270, 2008.</ref>

<ref name="Kue10">P. Kürschner, "[http://nbn-resolving.de/urn:nbn:de:bsz:ch1-201001082 Two-sided eigenvalue methods for modal approximation]”, Master’s thesis, Chemnitz University of Technology,
Department of Mathematics, Germany, 2010.</ref>

<ref name="FreRM11">F. D. Freitas, N. Martins, S. L. Varricchio, J. Rommes and F. C. Veliz, "[https://doi.org/10.1109/TPWRS.2011.2136442 Reduced-Order Transfer Matrices from RLC Network Descriptor Models of Electric Power Grids]” IEEE Transactions on Power Systems, 26(4): 1905--1916, 2011.</ref>

<ref name="Wat03">N. Watson and J. Arrillaga, "[http://dx.doi.org/10.1049/PBPO039E Power Systems Electromagnetic Transients Simulation]”, IET, London, UK, 2003.</ref>
</references>

==Contact==

[[User:Rommes|Joost Rommes]] 
[[User:kuerschner|Patrick Kürschner]] 
[http://www.nelsonmartins.com/ Nelson Martins] 
[http://www.ene.unb.br/index.php/component/professores/index.php?option=com_professores&view=professores&layout=perfil&id=131 Francisco D. Freitas]

Power system examples

2023-11-23T11:05:57Z

Malhotra: Undo revision 3853 by Malhotra (talk)

[[Category:benchmark]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:Differential_algebraic_system]]

__NUMBEREDHEADINGS__

==Description==

These first order systems are given in generalized state space form

:<math>
E\dot{x}(t)=A x(t)+B u(t), \quad
y(t)=Cx(t)+Du(t),\quad E,A\in\mathbb{R}^{n\times n},~B\in\mathbb{R}^{n\times m},~C\in\mathbb{R}^{p\times n},~D\in\mathbb{R}^{p\times m}\quad(1)
</math>

and originated at [http://www.cepel.br/ CEPEL] for simulating large '''power systems'''.

They come in different sizes and variants, including both SISO and MIMO systems having regular or singular <math>E</math> matrices. In the latter case the DAEs are of index 1 (except for PI Sections 20--80 which are index 2) and using simple row and column permutations, <math>E,A,B,C</math> can be brought into the form

:<math>
E=\left[ \begin{array}{cc}I_{n_f}&0\\0&0\end{array}\right],\quad A=\left[ \begin{array}{cc}A_{11}&A_{12}\\A_{21}&A_{22}\end{array}\right],\quad B=\left[ \begin{array}{cc}B_{1}\\B_2\end{array}\right],\quad C=\left[ \begin{array}{cc}C_{1}&C_2\end{array}\right],
</math>

where <math>n_f</math> denotes the number of finite eigenvalues in <math>\Lambda(A,E)</math> and <math>A_{22}\in\mathbb{R}^{n-n_f\times n-n_f}</math> is regular.
A complete overview over these systems can be found in table below. The '''power systems''' served as benchmark examples for [[Modal truncation|Dominant Pole based Modal Truncation]]<ref name="MarLP96"></ref><ref name="RomM06a"></ref><ref name="RomM06b"></ref><ref name="Rom07"></ref><ref name="RomM08"></ref><ref name="Kue10"></ref> and for a special adaption<ref name="FreRM08"></ref> of [[Balanced Truncation]] for the index-1 DAE systems.

==Data==
The table below lists the charateristics of all '''power systems'''. The files can be downloaded at [https://sites.google.com/site/rommes/software https://sites.google.com/site/rommes/software].

{| class="wikitable sortable" style="text-align: center; width: auto;"
|-
! Name
! <math>n</math>
! <math>m</math>
! <math>p</math>
! Type
|-
|New England
|66
|1
|1
|ODE
|-
|BIPS/97
|13251
|1
|1
|DAE
|-
|BIPS/1997
|13250
|1
|1
|DAE
|-
|BIPS/2007
|21476
|32
|32
|DAE
|-
|BIPS/97,MIMO8
|13309
|8
|8
|DAE
|-
|BIPS/97,MIMO28
|13251
|28
|28
|DAE
|-
|BIPS/97,MIMO46
|13250
|46
|46
|DAE
|-
|Juba5723
|40337
|2
|1
|DAE
|-
|Bauru5727
|40366
|2
|2
|DAE
|-
|zeros_nopss
|13296
|46
|46
|DAE
|-
|xingo6u
|20738
|1
|6
|DAE
|-
|nopss
|11685
|1
|1
|DAE
|-
|bips98_606
|7135
|4
|4
|DAE
|-
|bips98_1142
|9735
|4
|4
|DAE
|-
|bips98_1450
|11305
|4
|4
|DAE
|-
|bips07_1693
|13275
|4
|4
|DAE
|-
|bips07_1998
|15066
|4
|4
|DAE
|-
|bips07_2476
|16861
|4
|4
|DAE
|-
|bips07_3078
|21128
|4
|4
|DAE
|-
|PI Sections:
|
|
|
|
|-
|S10
|682
|1
|1
|DAE
|-
|S20
|1182
|1
|1
|DAE
|-
|S40
|2182
|1
|1
|DAE
|-
|S80
|4182
|1
|1
|DAE
|-
|M10
|682
|3
|3
|DAE
|-
|M20
|1182
|3
|3
|DAE
|-
|M40
|2182
|3
|3
|DAE
|-
|M80
|4182
|3
|3
|DAE
|-
|}

==Background==
Electrical '''power systems''' experience several steady-state and dynamic phenomena that may hinder its reliable, stable operation if not properly designed and operated. Among the dynamic phenomena, '''power system''' angle stability is of major concern, its studies requiring both numerical integration of a stiff set of large, nonlinear differential-algebraic (DAE) equations (for transient stability simulations) and the eigensolution as well as the use of other numerical linear algebra (NLA) algorithms applied to the linearized DAE equations of the same large stability models. '''Power systems''' with multiple electrical power plants, consumer loads, and industrial loads experience complex electromechanical oscillations, much as spring-mass mechanical systems experience mechanical oscillations. These oscillations, when the electric '''power system''' is under stressed conditions, may become poorly damped or unstable. The analysis of these low-frequency oscillations, as well as their controller-induced damping control, is enhanced by the results from NLA algorithms. The study of all these issues constitutes the field known as small signal stability.

There is a pressing need for better utilization of the transmission network and its cost-effective expansions to reliably carry electric power from the generating plants, driven by several primary energy sources, to the loads. This imposes stricter requirements on network design and '''power system''' control equipment, as well as on the adopted control laws, which, coupled with the continental dimensions of modern interconnected '''power systems''', must be studied with the help of simulators employing advanced NLA algorithms. This wiki page contains system models developed when dealing with these NLA issues, with emphasis on the analysis and control of small signal stability.

Modal analysis has been used for Model Order Reduction (MOR) of lightly damped systems like flexible mechanical structures and RLC networks from either '''power systems''' or microprocessor interconnectors. The advent of subspace accelerated dominant pole algorithms in <ref name="MarLP96"></ref><ref name="RomM06a"></ref><ref name="RomM06b"></ref>, made modal reduction effective for better damped systems, such as interconnected '''power systems''', which have eigenvalue clusters in the <math>0.2</math> to <math>2.0 Hz</math> range and damping ratios between <math>-0.05</math> and <math>0.25</math>. Important characteristics of model reduction methods include preserving the dominant dynamics and stability in the Reduced Order Models (ROMs).

Model reduction is important to the linear control system analysis and design of '''power system''' dynamic models. Reduced-order '''power system''' transfer function models are effective in lowering the computational costs of controller design. Once the dominant transfer function poles have been found, the calculation of their associated residues can be obtained by the scalar product of the left and right eigenvectors by the input and output vectors of the transfer function. From the knowledge of the dominant pole-residue set, one can build a transfer function ROM whose accuracy varies with the number of retained poles. See, e.g., the SADPA, SAMDP papers <ref name="RomM06a"></ref><ref name="RomM06b"></ref>.

Modal analysis is also used in the harmonic distortion analysis of '''power system''' networks, mainly in subtransmission and distribution voltage levels. In this application, modal analysis allows for determining the main network parameters that impact distortion levels and suggests changes to their effective values among other alternatives. The linear RLC network is modeled unloaded, the conventional electrical generators can be entirely neglected and the nonlinear loads of various nature are modeled as current injections into the network buses -3rd, 5th, 7th, 11th, 13th harmonics of the nominal system frequency (<math>50 Hz</math> or <math>60 Hz</math>). An even more important application for the modal analysis of these large descriptor system RLC network models is the production of ROM for multiport electrical networks, generally in the form of a state-space system whose dimension is considerably smaller than that of the original system. When these ROMs take the form of RLC networks, they are referred to as RLC reduced equivalents. These ROMs, or equivalents, are used in real-time and off-line simulators for the study of '''power system''' high-frequency transient phenomena. They are produced in varying degrees of complexity by using advanced NLA algorithms, which is an area of intense research work.

==Test systems for small-signal stability analysis of large electric power system networks==
All test '''power systems''' in [https://sites.google.com/site/rommes/software https://sites.google.com/site/rommes/software], except the PI sections example, are used for '''power system''' small-signal stability studies, and several of them are briefly described in <ref name="MarLP96"></ref>. Such studies include stability analysis, controller design, computation of MOR, etc. The dynamic phenomena of interest require the detailed modeling of the electrical energy generators (mainly the large sized ones) and other important devices such as flexible AC transmission systems (FACTS). The 3-phase alternating current (AC) network is comprised of nodes (electrical buses, which may be the representation of an entire electrical energy transmission (or distribution) substation, and branches (transmission lines, transformers, series capacitors) and need have only a static model in this application except for the occasional high-power electronic HVDC (High voltage direct current) transmission links and FACTS devices.

The input <math>u(t)</math> depends on the study of interest, but the excitation system voltage reference is a frequent input variable in small-signal stability studies. Another reference is the active power setpoint of a rotor-speed regulator. The input signal used in all the above mentioned files is of Vref type. The output could be an entry of the generalized state <math>x(t)</math> (state or algebraic variable) or a linear combination of these variables. The generalized states <math>x(t)</math> are associated with equations of generators, controllers and network. The interconnected network has only algebraic variables, except when FACTS devices are connected.
The test systems are represented by generalized state-space models, see <math>(1)</math>, where the feed through matrix <math>D</math> is generally zero.

==Test systems for electromagnetic transients and harmonic distortion studies==
Depending on the harmonic study at hand, a voltage or current source can be assigned as an input. In <ref name="MarLP96"></ref> the current injection <math>u(t)</math> at a node (also, electrical bus, in '''power system''' terminology) is the input, while the nodal voltage, at the same node, is the output (variable <math>y(t)</math>).

All transmission lines in the network are modeled by RLC ladder networks, of cascaded RLC PI-circuits, having fixed parameters <ref name="Wat03"></ref>. The transformers are modeled by series RL circuits. Loads and shunt elements are represented by series-connected RL (or RC) branch, or just an L (or C). As a consequence, the states <math>x(t)</math> of the dynamic system are either the current through an inductor or the voltage across a capacitor. So matrices <math>A, B, C, D, E</math> describe how each circuit element equation, node and voltage are connected, according to the Kirchhoff’s law.

On https://sites.google.com/site/rommes/software the systems labeled by PI Sections 20--80 are of index-2 ([https://drive.google.com/file/d/1ttJgCZeNEspA3Ty7vNPtsuPqo_-Qa-6g/view SISO_PI_n.zip and MIMO_PI_n.zip]). The data of these test systems can be converted into index-0 systems by applying a procedure for symbolic math elimination of algebraic variables and redundant state variables. This procedure is detailed in the Appendix C of <ref name="FreRM11"></ref>.

==References==
<references>
<ref name="MarLP96">N. Martins, L. Lima, and H. Pinto, "[https://doi.org/10.1109/59.486093 Computing dominant poles of power system transfer functions]", IEEE Transactions on
Power Systems, 11: 162--170, 1996.</ref>

<ref name="RomM06a">J. Rommes and N. Martins, "[https://doi.org/10.1109/TPWRS.2006.876671 Efficient computation of transfer function dominant poles using subspace acceleration]", IEEE Transactions on
Power Systems, 21(3): 1218--1226, 2006.</ref>

<ref name="RomM06b">J. Rommes and N. Martins, "[https://doi.org/10.1109/TPWRS.2006.881154 Efficient computation of multivariable transfer function dominant poles using subspace acceleration]", IEEE Transactions on
Power Systems, 21(4): 1471--1483, 2006.</ref>

<ref name="Rom07">J. Rommes, "[https://dspace.library.uu.nl/handle/1874/21787 Methods for eigenvalue problems with applications in model order reduction]", Ph.D. dissertation, Universiteit
Utrecht, 2007.</ref>

<ref name="RomM08">J. Rommes and N. Martins, "[https://doi.org/10.1137/070684562 Computing transfer function dominant poles of large second-order dynamical systems]" SIAM Journal on Scientific Computing, 30(4): 2137--2157, 2008.</ref>

<ref name="FreRM08">F. Freitas, J. Rommes, and N. Martins, "[https://doi.org/10.1109/TPWRS.2008.926693 Gramian-based reduction method applied to large sparse power system descriptor models]" IEEE Transactions on Power Systems, 23(3): 1258--1270, 2008.</ref>

<ref name="Kue10">P. Kürschner, "[http://nbn-resolving.de/urn:nbn:de:bsz:ch1-201001082 Two-sided eigenvalue methods for modal approximation]”, Master’s thesis, Chemnitz University of Technology,
Department of Mathematics, Germany, 2010.</ref>

<ref name="FreRM11">F. D. Freitas, N. Martins, S. L. Varricchio, J. Rommes and F. C. Veliz, "[https://doi.org/10.1109/TPWRS.2011.2136442 Reduced-Order Transfer Matrices from RLC Network Descriptor Models of Electric Power Grids]” IEEE Transactions on Power Systems, 26(4): 1905--1916, 2011.</ref>

<ref name="Wat03">N. Watson and J. Arrillaga, "[http://dx.doi.org/10.1049/PBPO039E Power Systems Electromagnetic Transients Simulation]”, IET, London, UK, 2003.</ref>
</references>

==Contact==

[[User:Rommes|Joost Rommes]] 
[[User:kuerschner|Patrick Kürschner]] 
[http://www.nelsonmartins.com/ Nelson Martins] 
[http://www.ene.unb.br/index.php/component/professores/index.php?option=com_professores&view=professores&layout=perfil&id=131 Francisco D. Freitas]

Power system examples

2023-11-23T11:03:45Z

Malhotra: Add infobox

[[Category:benchmark]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:Differential_algebraic_system]]

{{Infobox
|Title = Anemometer
|Benchmark ID =
* anemometer1Param_n29008m1q1
* anemometer3Param_n29008m1q1
|Category = misc
|System-Class = AP-LTI-FOS
|nstates = 29008
|ninputs = 1
|noutputs = 1
|nparameters =
* 2
* 5
|components = A, B, C, E
|License = NA
|Creator = [[User:Feng]]
|Editor =
* [[User:Feng]]
* [[User:Himpe]]
* [[User:Lnor]]
* [[User:Baur]]
* [[User:Will]]
* [[User:Lund]]
|Zenodo-link = NA
}}

__NUMBEREDHEADINGS__

==Description==

These first order systems are given in generalized state space form

:<math>
E\dot{x}(t)=A x(t)+B u(t), \quad
y(t)=Cx(t)+Du(t),\quad E,A\in\mathbb{R}^{n\times n},~B\in\mathbb{R}^{n\times m},~C\in\mathbb{R}^{p\times n},~D\in\mathbb{R}^{p\times m}\quad(1)
</math>

and originated at [http://www.cepel.br/ CEPEL] for simulating large '''power systems'''.

They come in different sizes and variants, including both SISO and MIMO systems having regular or singular <math>E</math> matrices. In the latter case the DAEs are of index 1 (except for PI Sections 20--80 which are index 2) and using simple row and column permutations, <math>E,A,B,C</math> can be brought into the form

:<math>
E=\left[ \begin{array}{cc}I_{n_f}&0\\0&0\end{array}\right],\quad A=\left[ \begin{array}{cc}A_{11}&A_{12}\\A_{21}&A_{22}\end{array}\right],\quad B=\left[ \begin{array}{cc}B_{1}\\B_2\end{array}\right],\quad C=\left[ \begin{array}{cc}C_{1}&C_2\end{array}\right],
</math>

where <math>n_f</math> denotes the number of finite eigenvalues in <math>\Lambda(A,E)</math> and <math>A_{22}\in\mathbb{R}^{n-n_f\times n-n_f}</math> is regular.
A complete overview over these systems can be found in table below. The '''power systems''' served as benchmark examples for [[Modal truncation|Dominant Pole based Modal Truncation]]<ref name="MarLP96"></ref><ref name="RomM06a"></ref><ref name="RomM06b"></ref><ref name="Rom07"></ref><ref name="RomM08"></ref><ref name="Kue10"></ref> and for a special adaption<ref name="FreRM08"></ref> of [[Balanced Truncation]] for the index-1 DAE systems.

==Data==
The table below lists the charateristics of all '''power systems'''. The files can be downloaded at [https://sites.google.com/site/rommes/software https://sites.google.com/site/rommes/software].

{| class="wikitable sortable" style="text-align: center; width: auto;"
|-
! Name
! <math>n</math>
! <math>m</math>
! <math>p</math>
! Type
|-
|New England
|66
|1
|1
|ODE
|-
|BIPS/97
|13251
|1
|1
|DAE
|-
|BIPS/1997
|13250
|1
|1
|DAE
|-
|BIPS/2007
|21476
|32
|32
|DAE
|-
|BIPS/97,MIMO8
|13309
|8
|8
|DAE
|-
|BIPS/97,MIMO28
|13251
|28
|28
|DAE
|-
|BIPS/97,MIMO46
|13250
|46
|46
|DAE
|-
|Juba5723
|40337
|2
|1
|DAE
|-
|Bauru5727
|40366
|2
|2
|DAE
|-
|zeros_nopss
|13296
|46
|46
|DAE
|-
|xingo6u
|20738
|1
|6
|DAE
|-
|nopss
|11685
|1
|1
|DAE
|-
|bips98_606
|7135
|4
|4
|DAE
|-
|bips98_1142
|9735
|4
|4
|DAE
|-
|bips98_1450
|11305
|4
|4
|DAE
|-
|bips07_1693
|13275
|4
|4
|DAE
|-
|bips07_1998
|15066
|4
|4
|DAE
|-
|bips07_2476
|16861
|4
|4
|DAE
|-
|bips07_3078
|21128
|4
|4
|DAE
|-
|PI Sections:
|
|
|
|
|-
|S10
|682
|1
|1
|DAE
|-
|S20
|1182
|1
|1
|DAE
|-
|S40
|2182
|1
|1
|DAE
|-
|S80
|4182
|1
|1
|DAE
|-
|M10
|682
|3
|3
|DAE
|-
|M20
|1182
|3
|3
|DAE
|-
|M40
|2182
|3
|3
|DAE
|-
|M80
|4182
|3
|3
|DAE
|-
|}

==Background==
Electrical '''power systems''' experience several steady-state and dynamic phenomena that may hinder its reliable, stable operation if not properly designed and operated. Among the dynamic phenomena, '''power system''' angle stability is of major concern, its studies requiring both numerical integration of a stiff set of large, nonlinear differential-algebraic (DAE) equations (for transient stability simulations) and the eigensolution as well as the use of other numerical linear algebra (NLA) algorithms applied to the linearized DAE equations of the same large stability models. '''Power systems''' with multiple electrical power plants, consumer loads, and industrial loads experience complex electromechanical oscillations, much as spring-mass mechanical systems experience mechanical oscillations. These oscillations, when the electric '''power system''' is under stressed conditions, may become poorly damped or unstable. The analysis of these low-frequency oscillations, as well as their controller-induced damping control, is enhanced by the results from NLA algorithms. The study of all these issues constitutes the field known as small signal stability.

There is a pressing need for better utilization of the transmission network and its cost-effective expansions to reliably carry electric power from the generating plants, driven by several primary energy sources, to the loads. This imposes stricter requirements on network design and '''power system''' control equipment, as well as on the adopted control laws, which, coupled with the continental dimensions of modern interconnected '''power systems''', must be studied with the help of simulators employing advanced NLA algorithms. This wiki page contains system models developed when dealing with these NLA issues, with emphasis on the analysis and control of small signal stability.

Modal analysis has been used for Model Order Reduction (MOR) of lightly damped systems like flexible mechanical structures and RLC networks from either '''power systems''' or microprocessor interconnectors. The advent of subspace accelerated dominant pole algorithms in <ref name="MarLP96"></ref><ref name="RomM06a"></ref><ref name="RomM06b"></ref>, made modal reduction effective for better damped systems, such as interconnected '''power systems''', which have eigenvalue clusters in the <math>0.2</math> to <math>2.0 Hz</math> range and damping ratios between <math>-0.05</math> and <math>0.25</math>. Important characteristics of model reduction methods include preserving the dominant dynamics and stability in the Reduced Order Models (ROMs).

Model reduction is important to the linear control system analysis and design of '''power system''' dynamic models. Reduced-order '''power system''' transfer function models are effective in lowering the computational costs of controller design. Once the dominant transfer function poles have been found, the calculation of their associated residues can be obtained by the scalar product of the left and right eigenvectors by the input and output vectors of the transfer function. From the knowledge of the dominant pole-residue set, one can build a transfer function ROM whose accuracy varies with the number of retained poles. See, e.g., the SADPA, SAMDP papers <ref name="RomM06a"></ref><ref name="RomM06b"></ref>.

Modal analysis is also used in the harmonic distortion analysis of '''power system''' networks, mainly in subtransmission and distribution voltage levels. In this application, modal analysis allows for determining the main network parameters that impact distortion levels and suggests changes to their effective values among other alternatives. The linear RLC network is modeled unloaded, the conventional electrical generators can be entirely neglected and the nonlinear loads of various nature are modeled as current injections into the network buses -3rd, 5th, 7th, 11th, 13th harmonics of the nominal system frequency (<math>50 Hz</math> or <math>60 Hz</math>). An even more important application for the modal analysis of these large descriptor system RLC network models is the production of ROM for multiport electrical networks, generally in the form of a state-space system whose dimension is considerably smaller than that of the original system. When these ROMs take the form of RLC networks, they are referred to as RLC reduced equivalents. These ROMs, or equivalents, are used in real-time and off-line simulators for the study of '''power system''' high-frequency transient phenomena. They are produced in varying degrees of complexity by using advanced NLA algorithms, which is an area of intense research work.

==Test systems for small-signal stability analysis of large electric power system networks==
All test '''power systems''' in [https://sites.google.com/site/rommes/software https://sites.google.com/site/rommes/software], except the PI sections example, are used for '''power system''' small-signal stability studies, and several of them are briefly described in <ref name="MarLP96"></ref>. Such studies include stability analysis, controller design, computation of MOR, etc. The dynamic phenomena of interest require the detailed modeling of the electrical energy generators (mainly the large sized ones) and other important devices such as flexible AC transmission systems (FACTS). The 3-phase alternating current (AC) network is comprised of nodes (electrical buses, which may be the representation of an entire electrical energy transmission (or distribution) substation, and branches (transmission lines, transformers, series capacitors) and need have only a static model in this application except for the occasional high-power electronic HVDC (High voltage direct current) transmission links and FACTS devices.

The input <math>u(t)</math> depends on the study of interest, but the excitation system voltage reference is a frequent input variable in small-signal stability studies. Another reference is the active power setpoint of a rotor-speed regulator. The input signal used in all the above mentioned files is of Vref type. The output could be an entry of the generalized state <math>x(t)</math> (state or algebraic variable) or a linear combination of these variables. The generalized states <math>x(t)</math> are associated with equations of generators, controllers and network. The interconnected network has only algebraic variables, except when FACTS devices are connected.
The test systems are represented by generalized state-space models, see <math>(1)</math>, where the feed through matrix <math>D</math> is generally zero.

==Test systems for electromagnetic transients and harmonic distortion studies==
Depending on the harmonic study at hand, a voltage or current source can be assigned as an input. In <ref name="MarLP96"></ref> the current injection <math>u(t)</math> at a node (also, electrical bus, in '''power system''' terminology) is the input, while the nodal voltage, at the same node, is the output (variable <math>y(t)</math>).

All transmission lines in the network are modeled by RLC ladder networks, of cascaded RLC PI-circuits, having fixed parameters <ref name="Wat03"></ref>. The transformers are modeled by series RL circuits. Loads and shunt elements are represented by series-connected RL (or RC) branch, or just an L (or C). As a consequence, the states <math>x(t)</math> of the dynamic system are either the current through an inductor or the voltage across a capacitor. So matrices <math>A, B, C, D, E</math> describe how each circuit element equation, node and voltage are connected, according to the Kirchhoff’s law.

On https://sites.google.com/site/rommes/software the systems labeled by PI Sections 20--80 are of index-2 ([https://drive.google.com/file/d/1ttJgCZeNEspA3Ty7vNPtsuPqo_-Qa-6g/view SISO_PI_n.zip and MIMO_PI_n.zip]). The data of these test systems can be converted into index-0 systems by applying a procedure for symbolic math elimination of algebraic variables and redundant state variables. This procedure is detailed in the Appendix C of <ref name="FreRM11"></ref>.

==References==
<references>
<ref name="MarLP96">N. Martins, L. Lima, and H. Pinto, "[https://doi.org/10.1109/59.486093 Computing dominant poles of power system transfer functions]", IEEE Transactions on
Power Systems, 11: 162--170, 1996.</ref>

<ref name="RomM06a">J. Rommes and N. Martins, "[https://doi.org/10.1109/TPWRS.2006.876671 Efficient computation of transfer function dominant poles using subspace acceleration]", IEEE Transactions on
Power Systems, 21(3): 1218--1226, 2006.</ref>

<ref name="RomM06b">J. Rommes and N. Martins, "[https://doi.org/10.1109/TPWRS.2006.881154 Efficient computation of multivariable transfer function dominant poles using subspace acceleration]", IEEE Transactions on
Power Systems, 21(4): 1471--1483, 2006.</ref>

<ref name="Rom07">J. Rommes, "[https://dspace.library.uu.nl/handle/1874/21787 Methods for eigenvalue problems with applications in model order reduction]", Ph.D. dissertation, Universiteit
Utrecht, 2007.</ref>

<ref name="RomM08">J. Rommes and N. Martins, "[https://doi.org/10.1137/070684562 Computing transfer function dominant poles of large second-order dynamical systems]" SIAM Journal on Scientific Computing, 30(4): 2137--2157, 2008.</ref>

<ref name="FreRM08">F. Freitas, J. Rommes, and N. Martins, "[https://doi.org/10.1109/TPWRS.2008.926693 Gramian-based reduction method applied to large sparse power system descriptor models]" IEEE Transactions on Power Systems, 23(3): 1258--1270, 2008.</ref>

<ref name="Kue10">P. Kürschner, "[http://nbn-resolving.de/urn:nbn:de:bsz:ch1-201001082 Two-sided eigenvalue methods for modal approximation]”, Master’s thesis, Chemnitz University of Technology,
Department of Mathematics, Germany, 2010.</ref>

<ref name="FreRM11">F. D. Freitas, N. Martins, S. L. Varricchio, J. Rommes and F. C. Veliz, "[https://doi.org/10.1109/TPWRS.2011.2136442 Reduced-Order Transfer Matrices from RLC Network Descriptor Models of Electric Power Grids]” IEEE Transactions on Power Systems, 26(4): 1905--1916, 2011.</ref>

<ref name="Wat03">N. Watson and J. Arrillaga, "[http://dx.doi.org/10.1049/PBPO039E Power Systems Electromagnetic Transients Simulation]”, IET, London, UK, 2003.</ref>
</references>

==Contact==

[[User:Rommes|Joost Rommes]] 
[[User:kuerschner|Patrick Kürschner]] 
[http://www.nelsonmartins.com/ Nelson Martins] 
[http://www.ene.unb.br/index.php/component/professores/index.php?option=com_professores&view=professores&layout=perfil&id=131 Francisco D. Freitas]