MOR Wiki - User contributions [en]

Battery pack

2026-03-03T07:40:28Z

Schuetz2:

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:sparse]]
[[Category:linear]]
[[Category:first differential order]]
[[Category:ODE]]
[[Category:MIMO]]

{{Infobox
|Title = Battery pack
|Benchmark ID = batteryPack_n151642m10q10
|Category = misc
|System-Class = AP-LTI-FOS
|nstates = 151642
|ninputs = 10
|noutputs = 10
|nparameters = 2
|components = A, B, C, E
|License = Creative Commons Attribution 4.0 International
|Creator = Lucas Kostetzer ([http://www.cadfem.de/ CADFEM])
|Editor = [[User:Vettermann]]
|Zenodo-link = https://zenodo.org/records/10820678
}}

== Motivation ==
Due to the increasing interest in electromobility detailed knowledge about the thermal system behavior
of complete '''battery packs''' becomes crucial <ref name="geppert"/>. Many aspects like power performance and aging characteristics depend on the temperature of the system. Thus the use of reduced models is important for a fast simulation of the system behavior over a long period of time.

== Description ==

[[File:Battery.png|480px|thumb|right|<caption>Model of the battery pack.</caption>]]

The '''battery pack''' consists of 10 cells and a water-cooled plate .
The model has been generated and meshed in ANSYS.
SOLID90 elements have been used for the finite element discretization of the battery cells.
The flow problem has been modeled by FLUID116 elements, representing a one-dimensional pipe flow. Thereby
SURF152 elements have been used to realize the convection between fluid and cooled plate.
Thermal radiation and convection between battery pack and environment are neglected.
The contact between the individual battery cells and between cells and cooled plate has been modeled with
the help of the elements CONTA174 and TARGE170.

The ambient temperature as well as the fluid temperature at the entrance of the pipes has the given value of 0°C.

== Data ==
The parametric system of order <math> n=151642 </math> with <math> m=10 </math> inputs and <math> q=10 </math> outputs is of the following form

:<math>
\begin{align}
E\dot{T} & = (A + p_1 A_1 + p_2 A_2)T + Bu \\
y & = CT
\end{align}
</math>

with

:{| style="text-align:left"
| <math> E \in \mathbb{R}^{n\times n} </math> || - capacity matrix
|-
| <math> A \in \mathbb{R}^{n \times n} </math> || - basic conductivity matrix
|-
| <math> A_1 \in \mathbb{R}^{n \times n} </math> || - part of conductivity matrix arising from convection
|-
| <math> A_2 \in \mathbb{R}^{n \times n} </math> || - part of conductivity matrix arising from heat flux
|-
| <math> B \in \mathbb{R}^{n\times m} </math> || - input map
|-
| <math> C \in \mathbb{R}^{q\times n} </math> || - output map
|-
| <math> T \in \mathbb{R}^n </math> ||- state vector (temperature)
|-
| <math> u \in \mathbb{R}^{m} </math> ||- input vector (heat generation rate of each battery cell)
|-
| <math> y \in \mathbb{R}^{q} </math> ||- output vector (temperature at the center of each battery cell)
|-
|}

where <math> p_1 </math> is the heat transfer coefficient and <math> p_2 </math> is the mass flow in the pipes (FLUID116 elements with load heat flux).

All matrices can be downloaded from [https://zenodo.org/records/10820678 Zenodo]. The '''battery pack''' is a benchmark for problems
containing unsymmetric matrices.

In the original model the values <math>p_1=500\tfrac{W}{m^2 K}</math> and <math>p_2=0.1\tfrac{W}{m^2}</math> for the parameters have been used.

== Origin ==
The model was created in ANSYS by Lucas Kostetzer ([http://www.cadfem.de/ CADFEM]). The export of the system matrices from ANSYS and the documentation for MOR Wiki were performed by [[User:Vettermann]].

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

For the benchmark itself and its data:

@misc{dataBatteryPack,
author = {Lucas Kostetzer and Julia Vettermann},
title = {Battery pack},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2024,
doi = {10.5281/zenodo.10820678}
}

== Reference ==
<references>
<ref name="geppert"> Geppert, Michael: "Entwicklung einer Werkzeugkette für die Erstellung reduzierter thermischer Modelle für Batteriepacks", diploma thesis, Institute of Automotive Technology, Technische Universität München, Munich, Germany (2010).</ref>
</references>

Machine tool MAX

2026-03-03T07:39:44Z

Schuetz2:

{{preliminary}} 

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

{{Infobox
|Title = Machine tool MAX
|Benchmark ID =
* machineToolMAX_feCoupled_n1265497m69q11
* machineToolMAX_outputCoupled_n1265497m287q224
|Category = CRC-TR-96
|System-Class = LTI-FOS
|nstates = 1265497
|ninputs =
* 69
* 287
|noutputs =
* 11
* 224
|nparameters = 0
|components = A, B, C, E
|License = Creative Commons Attribution 4.0 International
|Creator =
* CRC/TR 96:
* Chair of Machine Tools Development and Adaptive Controls (TU Dresden),
* Chair of Dynamics and Mechanism Design (TU Dresden),
* Mathematics in Industry and Technology (TU Chemnitz),
* Numerical Mathematics (Partial Differential Equations) (TU Chemnitz)
|Editor = [[User:Vettermann]]
|Zenodo-link = https://zenodo.org/records/10041041
}}

==Motivation==

Due to the increasing interest in manufacturing accuracy without an additional energy demand for cooling, knowledge about the thermo-elastic behavior of entire machine tools becomes crucial. Methods to correct the thermally induced position error between the tool-center-point (TCP) and the workpiece in real-time are needed. Therefore, reduced-order models that enable a fast simulation of entire machine tools are applied in the design and production process.

==Description==
<figure id="fig:plot1">
[[File:MAX_real.jpg|480px|thumb|right|<caption>Experimental machine tool MAX.</caption>]]
</figure>

<figure id="fig:plot2">
[[File:MAX_FE.pdf|480px|thumb|right|<caption>Subassemblies and FE-model of the benchmark MAX.</caption>]]
</figure>

The '''MAX''' benchmark is a linear time-invariant thermal model of the demonstrator machine tool '''[https://tu-dresden.de/ing/maschinenwesen/imd/lwm/forschung/werkzeugmaschinen-versuchsfeld/versuchstraeger-max MAX]'''(Fig. 1). Due to its lightweight construction based on aluminium structures, low heat capacities and high thermal coefficients of expansion are to be expected <ref name="grossmann2015"/>. It is divided into 50 stationary subassemblies (SA), see Fig. 2. The thermal finite element (FE) model was generated in ANSYS and afterwards the model was exported for post-processing as described in <ref name="vettermann2021"/>. The interaction between the subassemblies is modeled by contact boundary conditions. The evolution of the temperature field is modeled with the heat equation

:<math>
\begin{align}
c_p\rho\frac{\partial T}{\partial t}&=\lambda \Delta T, & &\text{ on } \Omega_k , \text{ } k=1,\dots,50, \\
\lambda\frac{\partial T}{\partial n}&=f, & &\text{ on } \Gamma_{c_k} \subset\partial\Omega_k , \text{ } k=1,\dots,50, \\
\lambda\frac{\partial T}{\partial n}&=\alpha_{ext}(T_{ext}-T), & &\text{ on } \Gamma_{ext_k} \subset\partial\Omega_k , \text{ } k=1,\dots,50, \\
T(0)&=T_0, & &\text{ at } t=0
\end{align}
</math>

with

:<math>
\begin{align}
f=\kappa_c(T_{c_i}-T_{c_k}), \quad i\neq k,
\end{align}
</math>

and

:{| style="text-align:left"
| <math> k </math> || - number of the subassembly, <math> k=1,\dots, 50 </math>
|-
| <math> T </math> || - temperature
|-
| <math> c_p </math> || - specific heat capacity
|-
| <math> \rho </math> || - density
|-
| <math> \lambda </math> || - heat conductivity
|-
| <math> \Omega_k </math> || - domain of the k-th subassembly
|-
| <math> \Gamma_{c_k} </math> || - contact boundary of the k-th subassembly (partly time varying, moves with the position of the z-slide (considered as heat flow))
|-
| <math> \kappa_{ext} </math> || - heat transfer coefficient between a subassembly and the ambient air
|-
| <math> T_{ext} </math> || - external temperature
|-
| <math> \Gamma_{ext_k} </math> || - contact boundary with the ambience
|-
| <math> \kappa_c </math> || - heat transfer coefficient between two subassemblies
|-
| <math> T_{c_k} </math> || - temperature of the contact area of subassembly k.
|-
|}

The finite element discretization of the heat conduction models leads to the system

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

with

:{| style="text-align:left"
| <math> E \in \mathbb{R}^{n\times n} </math> || - capacity matrix
|-
| <math> A \in \mathbb{R}^{n\times n} </math> || - conductivity matrix
|-
| <math> T \in \mathbb{R}^n </math> || - state vector (discrete temperature)
|-
| <math> B \in \mathbb{R}^{n \times m} </math> || - input map
|-
| <math> u \in \mathbb{R}^m </math> || - input vector
|-
| <math> y \in \mathbb{R}^p </math> || - output vector
|-
| <math> C \in \mathbb{R}^{p \times n} </math> || - output map.
|-
|}

==Data==

<figure id="fig:plot3">
[[File:coupled_systems.pdf|480px|thumb|right|<caption>Sketch of the two coupling approaches using the example of a model with 2 subassemblies.</caption>]]
</figure>

The system matrices are available in the .mat file format and can be downloaded from [https://zenodo.org/records/10041041 Zenodo]. The model has <math> n=1\,265\,497 </math> degrees of freedom and is available for two different coupling approaches, which are explained in more detail in <ref name="vettermann2021"/>, see also Fig. 3:
* A so called output-coupled model: In this case, an input-output coupling was used, i.e., the model is block diagonal with a total number of <math> m=287 </math> inputs and <math> p=224 </math> outputs. Thus the subassemlby models can be reduced separately for structure preserving MOR. The block diagonal system can be split into a set of matrices for each subassembly by the script "generate_subsystems.m".
* A so called FE-coupled model: In this case, the subassemblies are coupled on FE-level, i.e., the conductivity matrix A of the overall system has additional (off-diagonal) coupling blocks. Thus the subassembly models cannot be reduced separately anymore. There is one set of matrices for the overall model with <math> m=69 </math> inputs and <math> p=11 </math> outputs.

==Origin==

The '''MAX''' is used as an experimental machine tool in the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Category:CRC-TR-96 CRC/TR 96] Project-ID 174223256 financed by the German Research Foundation DFG.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:

@misc{dataCRCTR9623a,
author = {Collaborative Research Centre Transregio 96 (CRC/TR 96)},
title = {Machine tool model},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2023,
doi = {10.5281/zenodo.10041041}
}

* For the background on the benchmark:

@article{morVetSNetal21,
author = {Vettermann, J. and Sauerzapf, S. and Naumann, A. and Beitelschmidt, M. and Herzog, R. and Benner, P. and Saak, J.},
title = {Model order reduction methods for coupled machine tool models},
journal = {MM Science Journal},
volume = {Special Issue ICTIMT2021 --- 2nd International Conference on Thermal Issues in Machine Tools, April 20, 2021, Prague, Czech Republic},
number = 3,
pages = {4652--4659},
issn = {1805-0476},
year = {2021},
doi = {10.17973/MMSJ.2021_7_2021072}
}

==References==
<references>
<ref name="grossmann2015">
K. Großmann, ed., "[https://doi.org/10.1007/978-3-319-12625-8 Thermo-energetic Design of Machine Tools]", Springer International Publishing, Switzerland, pp. 9-10, 2015.
</ref>

<ref name="vettermann2021">
J. Vettermann, S. Sauerzapf, A. Naumann, M. Beitelschmidt, R. Herzog, P. Benner and J. Saak, "[https://doi.org/10.17973/MMSJ.2021_7_2021072 Model order reduction methods for coupled machine tool models]", MM Science Journal, Special Issue ICTIMT 2021, 2021.
</ref>

</references>

==Contact==

''[[User:Saak]]''

MAX stand

2026-03-03T07:39:23Z

Schuetz2:

{{preliminary}} 

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

{{Infobox
|Title = MAX stand
|Benchmark ID =
* maxStandSA1_n25872m10q10
* maxStandSA2_n39527m30q30
* maxStandSA3_n13551m6q6
* maxStandSA4_n4813m23q23
|Category = CRC-TR-96
|System-Class = AP-LTI-FOS
|nstates =
* 25872
* 39527
* 13551
* 4813
|ninputs =
* 10
* 30
* 6
* 23
|noutputs =
* 10
* 30
* 6
* 23
|nparameters =
* 2
* 1
* 1
* 1
|components = A, B, C, E
|License = Creative Commons Attribution 4.0 International
|Creator =
* CRC/TR 96:
* Chair of Machine Tools Development and Adaptive Controls (Technische Universität Dresden),
* Mathematics in Industry and Technology (Technische Universität Chemnitz)
|Editor = [[User:Vettermann]]
|Zenodo-link = https://zenodo.org/records/10039696
}}

==Motivation==

Due to the increasing interest in manufacturing accuracy without an additional energy demand for cooling, knowledge about the thermo-elastic behavior of entire machine tools becomes crucial. Methods to correct the thermally induced position error between the tool-center-point (TCP) and the workpiece in real-time are needed. Therefore, reduced-order models that enable a fast simulation of entire machine tools are applied in the design and production process.

==Description==
<figure id="fig:plot1">
[[File:MAX_stand.png|480px|thumb|right|<caption>Subassemblies of the MAX stand.</caption>]]
</figure>

The '''MAX stand''' benchmark is a linear thermal model of a stand of the demonstrator machine tool '''[https://tu-dresden.de/ing/maschinenwesen/imd/lwm/forschung/werkzeugmaschinen-versuchsfeld/versuchstraeger-max MAX]'''. Due to its lightweight construction based on aluminium structures, low heat capacities and high thermal coefficients of expansion are to be expected <ref name="grossmann2015"/>. It is divided into four stationary subassemblies (SA): base, z-stand with guide rail and bottom plate, motor with belt housing, bearing seat and outer bearing ring as well as ball screw (spindle with inner bearing ring), see Fig. 1. The thermal finite element (FE) model was generated in ANSYS and afterwards the FE system matrices were exported for post-processing, such as MOR, to MATLAB. The interaction between the subassemblies is modeled by contact boundary conditions. The evolution of the temperature field is modeled with the heat equation

:<math>
\begin{align}
c_p\rho\frac{\partial T}{\partial t}&=\lambda \Delta T, & &\text{ on } \Omega_k , \text{ } k=1,\dots,4, \\
\lambda\frac{\partial T}{\partial n}&=f, & &\text{ on } \Gamma_{c_k} \subset\partial\Omega_k , \text{ } k=1,\dots,4, \\
\lambda\frac{\partial T}{\partial n}&=\alpha_{ext}(T_{ext}-T), & &\text{ on } \Gamma_{ext_k} \subset\partial\Omega_k , \text{ } k=1,\dots,4, \\
T(0)&=T_0, & &\text{ at } t=0
\end{align}
</math>

with

:<math>
\begin{align}
f=q_{fric}+\kappa_c(T_{c_i}-T_{c_k}), \quad i,k=1,\dots,4, \quad i\neq k,
\end{align}
</math>

and

:{| style="text-align:left"
| <math> k </math> || - number of the subassembly, <math> k=1,\dots, 4 </math>
|-
| <math> T </math> || - temperature
|-
| <math> c_p </math> || - specific heat capacity
|-
| <math> \rho </math> || - density
|-
| <math> \lambda </math> || - heat conductivity
|-
| <math> \Omega_k </math> || - domain of the k-th subassembly
|-
| <math> \Gamma_{c_k} </math> || - contact boundary of the k-th subassembly (partly time varying, moves with the position of the z-slide (considered as heat flow))
|-
| <math> \kappa_{ext} </math> || - heat transfer coefficient between a subassembly and the ambient air
|-
| <math> T_{ext} </math> || - external temperature
|-
| <math> \Gamma_{ext_k} </math> || - contact boundary with the ambience
|-
| <math> q_{fric} </math> || - friction driven heat flow induced by the movement
|-
| <math> \kappa_c </math> || - heat transfer coefficient between two subassemblies
|-
| <math> T_{c_k} </math> || - temperature of the contact area of subassembly k.
|-
|}

The finite element discretization of the heat conduction models leads to the four systems

:<math>
\begin{align}
E_k\dot{T}_k&=(A_k-\sum_{i=1}^{l_k}\kappa_{ext_i}(t)A^i_k)T_k+B_k u_k(t),\quad k=1,\dots,4\\
y_k(t)&=C_k T_k,\\
\end{align}
</math>

with

:{| style="text-align:left"
| <math> E_k \in \mathbb{R}^{n\times n} </math> || - FE mass matrix
|-
| <math> A_k \in \mathbb{R}^{n\times n} </math> || - basic stiffness matrix (discrete Laplacian)
|-
| <math> l_k \in \mathbb{R} </math> || - number of boundary conditions with the ambience (Robin boundary conditions (RBC))
|-
| <math> A^i_k \in \mathbb{R}^{n\times n} </math> || - boundary mass matrix (part of stiffness matrix arising from RBC)
|-
| <math> T_k \in \mathbb{R}^n </math> || - state vector (discrete temperature)
|-
| <math> B_k \in \mathbb{R}^{n \times m} </math> || - input map
|-
| <math> u_k \in \mathbb{R}^m </math> || - input vector
|-
| <math> y_k \in \mathbb{R}^p </math> || - output vector
|-
| <math> C_k \in \mathbb{R}^{p \times n} </math> || - output map.
|-
|}

The movement of the z-slide is considered as position-dependent heat flows in the input <math> u(t) </math>. The position of the z-slide in the '''MAX stand''' model is characterized by the position <math> d \in [0, 0.7571]\text{ }m </math> of the z-slide on the spindle and the guide rail. To include this variability in the model the boundaries of the subassemblies
are divided into smaller parts and the average temperature of these parts is used for computing the input <math> u(t) </math>. To be more precise, the spindle and the guide rail are divided into 20 segments each. The segments are numbered from top to bottom, i.e. for the position <math>d=0</math> the z-slide is in contact with segment 1 of the spindle and the guide rail.
In the simulation the calculated average temperature of an area in the current time step serves as input temperature of this area in the next time step and is also used for calculating the thermally dependent heat transfer coefficients. Thus the average temperatures of all contact areas are needed as outputs <math> y(t) </math>. Further, the temperatures in certain nodes of interest complete the output <math> y(t) </math>. For more information concerning MOR for systems with moving loads see e.g. <ref name="lang2014"/> and the references therein.

==Data==

The system matrices (heat transfer coefficients with the ambience may be varied) and files for a simulation of the model in time domain can be downloaded from [https://zenodo.org/records/10039696 Zenodo]. There is a set of matrices for each subassembly. The numbering of the subassemblies is shown in Fig. 1. Further the archive contains a MATLAB file with the description and computation of the contacts between the subassemblies and a file for the simulation of a work process. The overall system dimension is <math> n=83\,763 </math> with

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Subassembly <math>k</math>
|<math>n_k</math>
|<math>m_k</math>
|<math>p_k</math>
|-style="background-color:#FFFFFF;"
|1
|<math>25\,872</math>
|<math>10</math>
|<math>10</math>
|-style="background-color:#FFFFFF;"
|2
|<math>39\,527</math>
|<math>30</math>
|<math>30</math>
|-style="background-color:#FFFFFF;"
|3
|<math>13\,551</math>
|<math>6</math>
|<math>6</math>
|-style="background-color:#FFFFFF;"
|4
|<math>4\,813</math>
|<math>23</math>
|<math>23</math>
|-
|}

==Remarks==
* This is a model with diagonal matrix <math>E</math>.
* The model was also used for numerical experiments presented in <ref name="palitta2023"/>.
* For the simulation the system was shifted by the initial temperature <math>T_0=20^{\circ}C</math> to guarantee a zero inital value.

==Origin==

The '''MAX stand''' is used as a demonstrator in the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Category:CRC-TR-96 CRC/TR 96] Project-ID 174223256 financed by the German Research Foundation DFG.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:

@misc{dataCRCTR9623,
author = {Collaborative Research Centre Transregio 96 (CRC/TR 96)},
title = {Model of a machine tool stand},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2023,
doi = {10.5281/zenodo.10039696}
}

==References==
<references>
<ref name="grossmann2015">
K. Großmann, ed., "[https://doi.org/10.1007/978-3-319-12625-8 Thermo-energetic Design of Machine Tools]", Springer International Publishing, Switzerland, pp. 9-10, 2015.
</ref>

<ref name="lang2014">
N. Lang, J. Saak and P. Benner, "[https://doi.org/10.1515/auto-2014-1095 Model Order Reduction for Systems with Moving Loads]", at-Automatisierungstechnik, pp. 512–522, 2014.
</ref>

<ref name="palitta2023">
P. Benner, D. Palitta and J. Saak, "[https://doi.org/10.1007/s11075-022-01409-5 On an integrated Krylov-ADI solver for large-scale Lyapunov equations]", Numer Algor 92, pp. 35–63, 2023.
</ref>
</references>

==Contact==

''[[User:Saak]]''

MiniHex

2026-03-03T07:39:11Z

Schuetz2:

{{preliminary}} 
[[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 = MiniHex
|Benchmark ID =
* MiniHexSA1_n44902m9q12
* MiniHexSA2_n49590m32q35
* MiniHexSA3_n1336m3q4
* MiniHexSA4_n21064m17q19
|Category = CRC-TR-96
|System-Class = AP-LTV-FOS
|nstates =
* 44902
* 49590
* 1336
* 21064
|ninputs =
* 9
* 32
* 3
* 17
|noutputs =
* 12
* 35
* 4
* 19
|nparameters =
* 6
* 2
* 1
* 9
|components = A, B, C, E
|License = Creative Commons Attribution 4.0 International
|Creator =
* CRC/TR 96:
* Chair of Machine Tools Development and Adaptive Controls (Technische Universität Dresden),
* Mathematics in Industry and Technology (Technische Universität Chemnitz)
|Editor = [[User:Vettermann]]
|Zenodo-link = https://zenodo.org/records/10033872
}}

==Motivation==

Due to the increasing interest in manufacturing accuracy without an additional energy demand for cooling, knowledge about the thermo-elastic behavior of entire machine tools becomes crucial. Methods to
correct the thermally induced position error between the tool-center-point (TCP) and the workpiece in real-time are needed. Therefore reduced-order models that enable a fast simulation of entire machine tools are
applied in the design and production process.

==Description==


[[File:MiniHex.png|480px|thumb|right|<caption>Figure 1: Thermal FE-modeling with segmented spindle surface to enter position-dependent heat flows <ref name="galant2015"/></caption>]]


The mobile demonstrator-machine '''[https://tu-dresden.de/ing/maschinenwesen/imd/lwm/forschung/werkzeugmaschinen-versuchsfeld/hexapod-minihex MiniHex]''' consists of six struts of variable length, which are driven by ball screws, each with a servo-motor <ref name="grossmann2015"/>. Following one of the feedaxes is considered. It is divided into four stationary
subassemblies: belt housing with motor, spindle, spindle nut and feed tube. The thermal finite element (FE) model was generated in ANSYS and afterwards the FE system matrices were exported for post-proceessing, such as MOR, to MATLAB.
The evolution of the temperature field is modelled with the heat equation

:<math>
\begin{align}
c_p\rho\frac{\partial T}{\partial t}&=\lambda \Delta T, & &\text{ on } \Omega_k , \text{ } k=1,\dots,4, \\
\lambda\frac{\partial T}{\partial n}&=f, & &\text{ on } \Gamma_{c_k} \subset\partial\Omega_k , \text{ } k=1,\dots,4, \\
\lambda\frac{\partial T}{\partial n}&=\kappa_{ext}(T_{ext}-T), & &\text{ on } \Gamma_{ext_k} \subset\partial\Omega_k , \text{ } k=1,\dots,4, \\
T(0)&=T_0, & &\text{ at } t=0
\end{align}
</math>

with

:<math>
\begin{align}
f=q_{fric}+\kappa_c(T_{c_i}-T_{c_k}), \quad i,k=1,\dots,4, \quad i\neq k,
\end{align}
</math>

and

:{| style="text-align:left"
| <math> k \in \mathbb{R} </math> || - number of the subassembly, <math> k=1,\dots, 4 </math>
|-
| <math> T </math> || - temperature
|-
| <math> c_p </math> || - specific heat capacity
|-
| <math> \rho </math> || - density
|-
| <math> \lambda </math> || - heat conductivity
|-
| <math> \Omega_k </math> || - domain of the k-th subassembly
|-
| <math> \Gamma_{c_k} </math> || - contact boundary of the k-th subassembly (partly time varying, moves with the position of the nut)
|-
| <math> \kappa_{ext} </math> || - heat transfer coefficient between a subassembly and the ambient air
|-
| <math> T_{ext} </math> || - external temperature
|-
| <math> \Gamma_{ext_k} </math> || - contact boundary with the ambience
|-
| <math> q_{fric} </math> || - friction driven heat flow induced by the movement
|-
| <math> \kappa_c </math> || - heat transfer coefficient between two subassemblies
|-
| <math> T_{c_k} </math> || - temperature of the contact area of subassembly k.
|-
|}

The finite element discretization of the heat conduction models leads to the four systems

:<math>
\begin{align}
E_k\dot{T}_k&=(A_k-\sum_{i=1}^{l_k}\kappa_{ext_i}(t)A^i_k)T_k+B_k u_k(t),\quad k=1,\dots,4\\
y_k(t)&=C_k T_k,\\
\end{align}
</math>

with

:{| style="text-align:left"
| <math> E_k \in \mathbb{R}^{n\times n} </math> || - FE mass matrix
|-
| <math> A_k \in \mathbb{R}^{n\times n} </math> || - basic stiffness matrix (discrete Laplacian)
|-
| <math> l \in \mathbb{R} </math> || - number of boundary conditions with the ambience (Robin boundary conditions (RBC))
|-
| <math> A^i_k \in \mathbb{R}^{n\times n} </math> || - boundary mass matrix (part of stiffness matrix arising from RBC)
|-
| <math> T_k \in \mathbb{R}^n </math> || - state vector (discrete temperature)
|-
| <math> B_k \in \mathbb{R}^{n \times m} </math> || - input map
|-
| <math> u_k \in \mathbb{R}^m </math> || - input vector
|-
| <math> y_k \in \mathbb{R}^p </math> || - output vector
|-
| <math> C_k \in \mathbb{R}^{p \times n} </math> || - output map.
|-
|}

The interaction between the subassemblies is considered as position-dependent heat flows in the input <math> u(t) </math>. The movement of the '''MiniHex''' is characterized by the position <math> d </math> of the nut on the spindle with
<math> d \in [-0.26,0.26]\text{ }m </math>. This movement induces a variability of the contact areas between the subassemblies. To include this variability in the model the boundaries of the subassemblies
are divided into smaller parts and the average temperature of these parts is used for computing the input <math> u(t) </math>. To be more precise, the spindle is divided into 32 parts and the feed tube into 17 parts <ref name="galant2016"/>.
In the simulation the calculated average temperature of an area in the current time step serves as input temperature of this area in the next time step and is also used for calculating the thermally dependent
heat transfer coefficients. Thus the average temperatures of all contact areas are needed as outputs <math> y(t) </math>. Further, the temperatures in certain nodes of interest complete the output <math> y(t) </math>.
For more information concerning MOR for systems with moving loads see e.g. <ref name="lang2014"/> and the references therein.

==Data==

The system matrices can be downloaded from [https://zenodo.org/records/10033872 Zenodo]. There is a set of matrices for each subassembly and input data to simulate a work process. The numbering of the subassemblies is as follows:
1 - belt housing with motor, 2 - spindle, 3 - spindle nut and 4 - feed tube.
Further the archive contains some MATLAB files with the description and computation of the 18 contacts between the subassemblies as well as the computation of the temperature-dependent heat transfer coefficients. The overall system dimension is <math> n=116892 </math> with <math> n_1=44902 </math> , <math> n_2=49590 </math>, <math> n_3=1336 </math> and <math> n_4=21064 </math>.

==Origin==

The '''MiniHex''' is used as a demonstrator in the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Category:CRC-TR-96 CRC/TR 96] Project-ID 174223256 financed by the German Research Foundation DFG.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:

@misc{dataCRCTR9623b,
author = {Collaborative Research Centre Transregio 96 (CRC/TR 96)},
title = {Model of a machine tool axis},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2023,
doi = {10.5281/zenodo.10033872}
}

* For the background on the benchmark:

@inbook{GalMG15,
author = {Galant, A. and Mühl, A. and Gro{\ss}mann, K.},
title = {Thermo-Elastic Simulation of Entire Machine Tool},
booktitle = {Thermo-energetic Design of Machine Tools},
editor = {Gro{\ss}mann, K.},
publisher = {Springer International Publishing, Switzerland},
pages = {69--84},
year = {2015},
doi = {10.1007/978-3-319-12625-8}
}

==References==
<references>
<ref name="galant2016">
A. Galant, S. Schroeder, B. Kauschinger and M. Beitelschmidt, "Erstellung und Abgleich eines strukturbasierten thermischen Modells der kugelgewindegetriebenen Vorschubachsen eines Hexapoden", in Thermo-Energetische Gestaltung von Werkzeugmaschinen, [https://d-nb.info/1082556157 Tagungsband 4. Kolloquium zum SFB/TR 96], C. Brecher, ed., RWTH Aachen, Aachen, 2016, pp. 15–32.
</ref>

<ref name="galant2015">
A. Galant, A. Mühl and K. Großmann, "Thermo-Elastic Simulation of Entire Machine Tool", in [https://doi.org/10.1007/978-3-319-12625-8 Thermo-energetic Design of Machine Tools], K. Großmann, ed., Springer International Publishing, Switzerland, 2015, pp. 69–84.
</ref>

<ref name="grossmann2015">
K. Großmann, ed., "[https://doi.org/10.1007/978-3-319-12625-8 Thermo-energetic Design of Machine Tools]", Springer International Publishing, Switzerland, 2015, pp. 9-10.
</ref>

<ref name="lang2014">
N. Lang, J. Saak and P. Benner, "[https://doi.org/10.1515/auto-2014-1095 Model Order Reduction for Systems with Moving Loads]", at-Automatisierungstechnik, 8/2014, pp. 512–522.
</ref>
</references>

==Contact==

''[[User:Saak]]''

Simplified machine tool

2026-03-03T07:37:01Z

Schuetz2:

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

{{Infobox
|Title = Simplified machine tool
|Benchmark ID =
* Coarse:
* simplifiedMachineToolCoarseSA1_n28183m6q5
* simplifiedMachineToolCoarseSA2_n10080m8q8
* simplifiedMachineToolCoarseSA3_n3721m8q8
* simplifiedMachineToolCoarseSA4_n3276m5q5
* Fine:
* simplifiedMachineToolFineSA1_n106641m6q5
* simplifiedMachineToolFineSA2_n35408m8q8
* simplifiedMachineToolFineSA3_n11956m8q8
* simplifiedMachineToolFineSA4_n12812m5q5
|Category = CRC-TR-96
|System-Class = LTI-FOS
|nstates =
* Coarse:
* 28183
* 10080
* 3721
* 3276
* Fine:
* 106641
* 35408
* 11956
* 12812
|ninputs =
* 6
* 8
* 8
* 5
|noutputs =
* 5
* 8
* 8
* 5
|nparameters = 0
|components = A, B, C, E
|License = Creative Commons Attribution 4.0 International
|Creator =
* CRC/TR 96:
* Stefan Sauerzapf
* Andreas Naumann
* [[User:Vettermann]]
* [[User:Saak]]
|Editor = [[User:Vettermann]]
|Zenodo-link = https://doi.org/10.5281/zenodo.10017861
}}

==Motivation==

Due to the increasing interest in manufacturing accuracy without an additional energy demand for cooling, knowledge about the thermo-elastic behavior of entire machine tools becomes crucial. Methods to correct the thermally induced position error between the tool-center-point (TCP) and the workpiece in real-time are needed. Therefore, reduced-order models that enable a fast simulation of entire machine tools are applied in the design and production process.

==Description==
<figure id="fig:plot1">
[[File:DMU_Dummy_subassemblies.png|480px|thumb|right|<caption>Subassemblies of the simplified machine tool.</caption>]]
</figure>

The '''simplified machine tool''' benchmark is a linear time-invariant thermal model used for investigation and demonstration purposes in the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Category:CRC-TR-96 CRC/TR 96]. For more information on first numerical experiments, see <ref name="euspen2020"/>. It is divided into four stationary subassemblies (SA): machine bed, X-slide, Z-slide, Y-slide, see Fig. 1. The thermal finite element (FE) model was generated in ANSYS and afterwards the FE system matrices were exported for post-processing, such as MOR, to MATLAB. The interaction between the subassemblies is modeled by contact boundary conditions. In this case an input-output coupling was used, i.e., the model is block diagonal and thus the subassemly models can be reduced separately for structure preserving MOR. The evolution of the temperature field is modeled by the heat equation

:<math>
\begin{align}
c_p\rho\frac{\partial T}{\partial t}&=\lambda \Delta T, & &\text{ on } \Omega_k , \text{ } k=1,\dots,4, \\
\lambda\frac{\partial T}{\partial n}&=f, & &\text{ on } \Gamma_{c_k} \subset\partial\Omega_k , \text{ } k=1,\dots,4, \\
\lambda\frac{\partial T}{\partial n}&=\alpha_{ext}(T_{ext}-T), & &\text{ on } \Gamma_{ext_k} \subset\partial\Omega_k , \text{ } k=1,\dots,4, \\
T(0)&=T_0, & &\text{ at } t=0
\end{align}
</math>

with

:<math>
\begin{align}
f=q_{fric}+\kappa_c(T_{c_i}-T_{c_k}), \quad i,k=1,\dots,4, \quad i\neq k,
\end{align}
</math>

and

:{| style="text-align:left"
| <math> k </math> || - number of the subassembly, <math> k=1,\dots, 4 </math>
|-
| <math> T </math> || - temperature
|-
| <math> c_p </math> || - specific heat capacity
|-
| <math> \rho </math> || - density
|-
| <math> \lambda </math> || - heat conductivity
|-
| <math> \Omega_k </math> || - domain of the k-th subassembly
|-
| <math> \Gamma_{c_k} </math> || - contact boundary of the k-th subassembly (partly time varying, moves with the position of the z-slide (considered as heat flow))
|-
| <math> \kappa_{ext} </math> || - heat transfer coefficient between a subassembly and the ambient air
|-
| <math> T_{ext} </math> || - external temperature
|-
| <math> \Gamma_{ext_k} </math> || - contact boundary with the ambience
|-
| <math> q_{fric} </math> || - friction driven heat flow induced by the movement
|-
| <math> \kappa_c </math> || - heat transfer coefficient between two subassemblies
|-
| <math> T_{c_k} </math> || - temperature of the contact area of subassembly k.
|-
|}

The finite element discretization of the heat conduction models leads to the four systems

:<math>
\begin{align}
E_k\dot{T}_k&=A_kT_k + B_k u_k(t),\quad k=1,\dots,4\\
y_k(t)&=C_k T_k,\\
\end{align}
</math>

with

:{| style="text-align:left"
| <math> E_k \in \mathbb{R}^{n\times n} </math> || - capacity matrix
|-
| <math> A_k \in \mathbb{R}^{n\times n} </math> || - conductivity matrix
|-
| <math> T_k \in \mathbb{R}^n </math> || - discrete temperature
|-
| <math> B_k \in \mathbb{R}^{n \times m} </math> || - input map
|-
| <math> u_k \in \mathbb{R}^m </math> || - input vector
|-
| <math> y_k \in \mathbb{R}^p </math> || - output vector
|-
| <math> C_k \in \mathbb{R}^{p \times n} </math> || - output map.
|-
|}

==Data==

The system matrices can be downloaded from [https://doi.org/10.5281/zenodo.10017861 zenodo]. There is a set of matrices for each subassembly. The numbering of the subassemblies is shown in Fig. 1. The model is available in two versions: a fine discretization with <math> n=166\,817 </math> degrees of freedom and a coarse discretization with an overall system dimension of <math> n=45\,260 </math>. The dimensions of the subassemblies of both models can be seen in the following table:

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Model
|Subassembly <math>k</math>
|<math>n_k</math>
|<math>m_k</math>
|<math>p_k</math>
|-style="background-color:#FFFFFF;"
|fine
|1
|<math>106\,641</math>
|<math>6</math>
|<math>5</math>
|-style="background-color:#FFFFFF;"
|
|2
|<math>35\,408</math>
|<math>8</math>
|<math>8</math>
|-style="background-color:#FFFFFF;"
|
|3
|<math>11\,956</math>
|<math>8</math>
|<math>8</math>
|-style="background-color:#FFFFFF;"
|
|4
|<math>12\,812</math>
|<math>5</math>
|<math>5</math>
|-style="background-color:#FFFFFF;"
|coarse
|1
|<math>28\,183</math>
|<math>6</math>
|<math>5</math>
|-style="background-color:#FFFFFF;"
|
|2
|<math>10\,080</math>
|<math>8</math>
|<math>8</math>
|-style="background-color:#FFFFFF;"
|
|3
|<math>3\,721</math>
|<math>8</math>
|<math>8</math>
|-style="background-color:#FFFFFF;"
|
|4
|<math>3\,276</math>
|<math>5</math>
|<math>5</math>
|-
|}

==Origin==

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

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:

@misc{dataSauNVetal23,
author = {S. Sauerzapf and A. Naumann and J. Vettermann and J. Saak},
title = {Matrices for a simplified machine tool model},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2023,
doi = {10.5281/zenodo.10017861}
}

* For the background on the benchmark:

@inproceedings{SauVNetal20,
author = {Sauerzapf, S. and Vettermann, J. and Naumann, A. and Saak, J. and Beitelschmidt, M. and Benner, P.},
title = {Simulation of the thermal behavior of machine tools for efficient machine development and
online correction of the {Tool} {Center} {Point} ({TCP})-displacement},
booktitle = {Conference Proceedings on Thermal Issues, Aachen, 26-27 February},
organization = {euspen},
pages = {135--138},
year = {2020},
url = {https://www.euspen.eu/knowledge-base/TI20125.pdf }
}

==References==

<references>
<ref name="euspen2020">
S. Sauerzapf, J. Vettermann, A. Naumann, J. Saak, M. Beitelschmidt and P. Benner, "[https://www.euspen.eu/knowledge-base/TI20125.pdf Simulation of the thermal behavior of machine tools for efficient machine development and online correction of the Tool Center Point (TCP)-displacement]", Conference Proceedings on Thermal Issues, Aachen, 26-27 February, pp. 135–138, euspen, 2020.
</ref>
</references>

==Contact==

''[[User:Saak]]''

Vertical Stand

2026-03-03T07:35:40Z

Schuetz2:

[[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
}}

==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. https://modelreduction.org/morwiki/Vertical_Stand

@MISC{morwiki_vertstand,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Vertical Stand},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/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]]''

Power system examples

2026-02-26T12:27:25Z

Schuetz2:

[[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
}}

==Description==

These first order systems are given in generalized state space form

:<math>
\begin{array}{rcl}
E\dot{x}(t)&=&A x(t)+B u(t), \\
y(t)&=&Cx(t)+Du(t),
\end{array}
</math>
where <math>E,A\in\mathbb{R}^{n\times n}</math>, <math>B\in\mathbb{R}^{n\times m}</math>, <math>C\in\mathbb{R}^{p\times n}</math>, <math>D\in\mathbb{R}^{p\times m}</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 \mathrm{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 \mathrm{Hz}</math> or <math>60 \mathrm{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 consists 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 to 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 above, 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]

Feed axis

2026-02-23T11:47:50Z

Schuetz2:

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

==Description==
<figure id="fig:plot1">
[[File:feed_axis.png|480px|thumb|right|<caption>Feed axis of a modern 5-axis vertical milling machine<ref name="vettermann22"/>.</caption>]]
</figure>

The '''Feed axis''' benchmark is a thermo-mechanical model of a subassembly of a modern 5-axis vertical milling machine described in<ref name="vettermann22"/>. Due to trade secrets the model provided here uses a coarser discretization and sliqhtly different material properties. The system consists of two large structural parts, namely the slide and the headstock, and four machine components which are the carriages, see Figure 1. During the finite element analysis, the parts are modeled as linear elements, while the components are considered as nonlinear spring-damper elements using multi-point constraints. The temperature change in the guideway systems of a machine tool can lead to a measurable change in the static stiffness and causes a nonlinearity in the model. Thus, the stiffness matrix has a small nonlinear part reflecting these geometrically local nonlinearities.

==Dimensions==
System structure:

:<math>
\begin{align}
E \dot{x}(t) + K 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>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=79\,814</math>, <math>m=20</math> and <math>p=63</math>.

==Data==
The data is available at [https://doi.org/10.5281/zenodo.6778458 Zenodo]. It contains information for 6 different poses of the model which can be combined in form of a switched system to consider a relative movement between slide and headstock.

==Remarks==
* The finite element discretization has been performed with [https://www.ansys.com Ansys].
* The system consists of a large linear part and a small nonlinear subsystem. A strategy for MOR by substructuring allowing the usage of standard methods for LTI systems is presented in<ref name="vettermann22"/>.
* It is a differential-algebraic system with one-sided coupling.
* Further investigations on MOR for this benchmark including the problem of moving loads modeled in form of a switched system can be found in<ref name="vettermann23"/>. Here, the scaling of the model plays an important role.
* The data package<ref name="datavettermann22"/> also provides codes for MOR and simulation of the model.

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

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

* For the benchmark itself and its data:
@misc{dataVS22,
author = {J. Vettermann and A. Steinert and J. Saak},
title = {Feed axis},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2022,
doi = {10.5281/zenodo.6778458}
}

* For the background on the benchmark:

@article{VS22,
author = {J. Vettermann and A. Steinert and C. Brecher and P. Benner and J. Saak},
title = {Compact thermo-mechanical models for the fast simulation of machine tools with nonlinear component behavior},
journal = {at - Automatisierungstechnik},
volume = 70,
number = 8,
year = 2022,
pages = {692--704},
doi = {10.1515/auto-2022-0029},
}

==References==

<references>

<ref name="vettermann22">J. Vettermann, A. Steinert, C. Brecher, P. Benner, J. Saak. "[https://doi.org/10.1515/auto-2022-0029 Compact thermo-mechanical models for the fast simulation of machinetools with nonlinear component behavior]", at - Automatisierungstechnik, 70(8): 692–704, 2022.</ref>

<ref name="datavettermann22">J. Vettermann, A. Steinert, J. Saak. [https://doi.org/10.5281/zenodo.6778458 Code and Data for Numerical Experiments in "Compact Thermo-Mechanical Models for the Fast Simulation of Machine Tools with Nonlinear Component Behavior"], zenodo, 2022.</ref>

<ref name="vettermann23">Q. Aumann, P. Benner, J. Saak, J. Vettermann. "[https://doi.org/10.1002/pamm.202200286 Model order reduction via substructuring for a nonlinear, differential-algebraic machine tool model with moving loads]", Proceedings in Applied Mathematics and Mechanics, 93(1), 2023.</ref>

</references>

==Contact==

''[[User:Saak]]''

RLC Ladder (Gugercin/Antoulas 2004)

2026-01-26T13:34:29Z

Schuetz2:

{{preliminary}} 

[[Category:benchmark]]
[[Category:SISO]]

{{Infobox
|Title = RCL Circuit Equations
|Benchmark ID =
* rlc_ladder_GA_n?m1q1
|Category = morwiki
|System-Class = LTI-FOS
|nstates =
* arbitrary
|ninputs =
* 1
|noutputs =
* 1
|nparameters = 0
|components = A, B, C, D
|License = MIT
|Creator = [[User:Gugercin]]
|Editor =
* [[User:Saak]]
|Zenodo-link = NA
}}

==Description==
<figure id="fig1">[[File:Rlc_ladder_segment_GugA04.png|490px|thumb|right|<caption>One section of the circuit.</caption>]]</figure>

This model represents an electic circuit made of a cascade of interconnected sections. Each section consists of a simple setup as depicted in [[RLC_Ladder_(Gugercin/Antoulas_2004)#fig1|Figure 1]]. The input is the voltage <math>V</math> applied to the first section; the output is the current <math>I</math> of the first section.

:<math>
\begin{align}
\dot x(t) &= A x(t) + B u(t), \\
y(t) &= C x(t) + D u(t), \\
x(0) &= x_0,
\end{align}
</math>

with <math>t>0</math>.

==Origin==
This model was used in the numerical experiments section of <ref name="morGugA04"/>.

==Data==
The following Matlab code assembles the above described <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>, matrices of requested dimensions. The input parameters <code>R</code>, <code>Rbar</code>, <code>Cap</code> and <code>L</code> represent the two resistances, the capacitance and the inductivity of each section of the circuit, as depicted in [[RLC_Ladder_(Gugercin/Antoulas_2004)#fig1|Figure 1]]. They default to the values from <ref name="morGugA04"/>, when omitted.

<div class="thumbinner" style="width:680px;text-align:left;">
<syntaxhighlight lang="matlab">
function [A, B, C, D, sys_rlc] = circuit_sparse(N, R , Rbar, Cap, L)
% circuit_sparse generates a SISO RLC ladder model of prescribed dimensions
%
% The model follows the descriptioin in DOI:10.1080/00207170410001713448
% and the input parameters R, Rbar, Cap and L default to the values given
% there, when omitted. If also N is omitted the original 100 dimensional
% realization is created.
%
% The code is a sparse adaption of the original dense generator function by
% Serkan Gugercin.
%

%
% MIT License
%
% Copyright (c) 2026 The MORwiki Community
%
% Permission is hereby granted, free of charge, to any person
% obtaining a copy of this software and associated documentation files
% (the "Software"), to deal in the Software without restriction,
% including without limitation the rights to use, copy, modify, merge,
% publish, distribute, sublicense, and/or sell copies of the Software,
% and to permit persons to whom the Software is furnished to do so,
% subject to the following conditions:
%
% The above copyright notice and this permission notice shall be
% included in all copies or substantial portions of the Software.
%
% THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
% EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
% MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
% NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS
% BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN
% ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
% CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
% SOFTWARE.
%

if nargin < 1
N = 100;
end

if nargin < 2
% parameter values from DOI:10.1080/00207170410001713448
R = 0.1;
Rbar = 1;
Cap = 0.1;
L = 0.1;
end

z1 = 1 / L * ones(N, 1);
z2 = -1 / L * Rbar * R / (Rbar + R) * ones(N, 1);
z3 = -1 / L * Rbar / (Rbar + R) * ones(N, 1);

% Every second row needs to be adapted for the second ladder step
z3(2:2:N) = -1 / Cap;
z2(1:2:N) = -1 / Cap / (Rbar + R);
z1(2:2:N) = 1 / Cap * Rbar / (Rbar + R);

% start end end are exceptional
z2(1) = -1 / R / Cap; % sets A(1, 1)
z3(1) = -1 / Cap; % sets A(1, 2)
z2(N) = -Rbar / L; % sets A(N, N)
z1(N - 1) = 1 / L; % sets A(N, N-1)

A = spdiags([z1 z2 z3], -1:1, N, N);
B = [(1 / R / Cap); sparse(N - 1, 1)];
C = [(-1 / R) sparse(1, N - 1)];
D = 1 / R;

if exist('sparss', 'file')
sys_rlc = sparss(A, B, C, D);
else
sys_rlc = [];
end
</syntaxhighlight>
</div>

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

* For the benchmark itself and its data:
:: '''RLC Ladder (Gugercin/Antoulas 2004)'''. hosted at MORwiki - Model Order Reduction Wiki, 2026. http://modelreduction.org/index.php/RLC_Ladder_(Gugercin/Antoulas_2004)

@MISC{morwiki_steel,
author = <nowiki>{{The MORWiki Community}}</nowiki>,
title = {RLC Ladder (Gugercin/Antoulas 2004)}
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/RLC_Ladder_(Gugercin/Antoulas_2004)}</nowiki>,
year = 2026
}

* For the background on the benchmark:

@Article{morGugA04,
author = {Gugercin, S. and Antoulas, A.~C.},
title = {A survey of model reduction by balanced truncation and some
new results},
journal = IntControl,
year = 2004,
volume = 77,
number = 8,
pages = {748--766},
doi = {10.1080/00207170410001713448}
}

==References==

<references>

<ref name="morGugA04"> S. Gugercin and A.C. Antoulas, [https://doi.org/10.1080/00207170410001713448 A survey of model reduction by balanced truncation and some new results], International Journal of Control,
77:8, 748-766, 2004.</ref>

</references>

Windscreen

2025-06-17T05:42:34Z

Schuetz2:

[[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. https://modelreduction.org/morwiki/Windscreen

@MISC{morwiki_windscreen,
author = <nowiki>{{Oberwolfach Benchmark Collection}}</nowiki>,
title = {Windscreen},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/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

2025-06-17T05:42:17Z

Schuetz2:

[[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. https://modelreduction.org/morwiki/Vertical_Stand

@MISC{morwiki_vertstand,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Vertical Stand},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/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

2025-06-17T05:41:53Z

Schuetz2:

[[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>{https://modelreduction.org/morwiki/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>

Thermal Model

2025-06-17T05:41:26Z

Schuetz2:

[[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. https://modelreduction.org/morwiki/Thermal_Model

@MISC{morwiki_thermal,
author = <nowiki>{{Oberwolfach Benchmark Collection}}</nowiki>,
title = {Thermal Model},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/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

2025-06-17T05:41:00Z

Schuetz2:

[[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. https://modelreduction.org/morwiki/Thermal_Block

@MISC{morwiki_thermalblock,
author = <nowiki>{Rave, S. and Saak, J.}</nowiki>,
title = {Thermal Block},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/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]]''

Synthetic parametric model

2025-06-17T05:40:39Z

Schuetz2:

[[Category:benchmark]]
[[Category:procedural]]
[[Category:parametric]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:Parametric]]
[[Category:first differential order]]
[[Category:SISO]]
[[Category:Sparse]]

==Description==

<figure id="fig1">[[File:synth_poles.png|600px|thumb|right|<caption>System poles for different parameter values.</caption>]]</figure>

On this page you will find a synthetic parametric model with one parameter for which one can easily experiment with different system orders, values of the parameter, as well as different poles and residues (see Fig. 1).
Also, the decay of the Hankel singular values can be changed indirectly through the parameter.

===Model===

We consider a dynamical system in the frequency domain given by its pole-residue form of the transfer function

:<math>
\begin{align}
H(s,\varepsilon) & = \sum_{k=1}^{N}\frac{r_{k}}{s-p_{k}}\\
& = \sum_{k=1}^{N}\frac{r_{k}}{s-(\varepsilon a_{k} + jb_{k})},
\end{align}
</math>

with <math>p_{k} = \varepsilon a_{k} + jb_{k}</math> the poles of the system, <math>j</math> the imaginary unit, and <math>r_{k}</math> the residues.
The parameter <math>\varepsilon</math> is used to scale the real part of the system poles.
We can write down the state-space realization of the system's transfer function as

:<math>
\begin{align}
H(s,\varepsilon) = \widehat{C}(sI_{N} - (\varepsilon \widehat{A}_{\varepsilon} + \widehat{A}_{0}))^{-1}\widehat{B},
\end{align}
</math>

with the corresponding system matrices <math>\widehat{A}_{\varepsilon} \in \mathbb{R}^{N \times N}</math>, <math>\widehat{A}_{0} \in \mathbb{C}^{N \times N}</math>, <math>\widehat{B} \in \mathbb{R}^{N}</math>, and <math>\widehat{C}^{T} \in \mathbb{C}^{N}</math> given by

:<math>
\begin{align}
\varepsilon\widehat{A}_{\varepsilon} + \widehat{A}_{0}
& = \varepsilon \begin{bmatrix} a_{1} & & \\ & \ddots & \\ & & a_{N} \end{bmatrix}
+ \begin{bmatrix} jb_{1} & & \\ & \ddots & \\ & & jb_{N} \end{bmatrix},\\
\widehat{B} & = \begin{bmatrix}1, & \ldots, & 1 \end{bmatrix}^{T},\\
\widehat{C} & = \begin{bmatrix}r_{1}, & \ldots, & r_{n} \end{bmatrix}.
\end{align}
</math>

One notices that the system matrices <math>\widehat{A}_{0}</math> and <math>\widehat{C}</math> have complex entries.
For rewriting the system with real matrices, we assume that <math>N</math> is even, <math>N=2m</math>, and that all system poles are complex and ordered in complex conjugate pairs, i.e.,

:<math>
\begin{align}
p_{1} & = \varepsilon a_{1} + jb_{1},\\
p_{2} & = \varepsilon a_{1} - jb_{1},\\
& \ldots\\
p_{N-1} & = \varepsilon a_{m} + jb_{m},\\
p_{N} & = \varepsilon a_{m} - jb_{m}.
\end{align}
</math>

Corresponding to the system poles, also the residues are written in complex conjugate pairs

:<math>
\begin{align}
r_{1} & = c_{1} + jd_{1},\\
r_{2} & = c_{1} - jd_{1},\\
& \ldots\\
r_{N-1} & = c_{m} + jd_{m},\\
r_N & = c_{m} - jd_{m}.
\end{align}
</math>

Using this, the realization of the dynamical system can be written with matrices having real entries by

:<math>
\begin{align}
A_{\varepsilon} & = \begin{bmatrix} A_{\varepsilon, 1} & & \\ & \ddots & \\ & & A_{\varepsilon, m} \end{bmatrix}, &
A_{0} & = \begin{bmatrix} A_{0, 1} & & \\ & \ddots & \\ & & A_{0, m} \end{bmatrix}, &
B & = \begin{bmatrix} B_{1} \\ \vdots \\ B_{m} \end{bmatrix}, &
C & = \begin{bmatrix} C_{1}, & \cdots, & C_{m} \end{bmatrix},
\end{align}
</math>

with <math>A_{\varepsilon, k} = \begin{bmatrix} a_{k} & 0 \\ 0 & a_{k} \end{bmatrix}</math>, <math>A_{0, k} = \begin{bmatrix} 0 & b_{k} \\ -b_{k} & 0 \end{bmatrix}</math>, <math>B_{k} = \begin{bmatrix} 2 \\ 0 \end{bmatrix}</math>, <math>C_{k} = \begin{bmatrix} c_{k}, & d_{k} \end{bmatrix}</math>.

<figure id="fig2">[[File:synth_freq_resp.png|600px|thumb|right|<caption>Frequency response of synthetic parametric system for different parameter values.</caption>]]</figure>

===Numerical Values===

<figure id="fig3">[[File:synth_hsv.png|600px|thumb|right|<caption>Hankel singular values of synthetic parametric system for different parameter values.</caption>]]</figure>

We construct a system of order <math>N = 100</math>.
The numerical values for the different variables are

* <math>a_{k}</math> equally spaced in the interval <math>[-10^3, -10]</math>,

* <math>b_{k}</math> equally spaced in the interval <math>[10, 10^3]</math>,

* <math>c_{k} = 1</math>,

* <math>d_{k} = 0</math>,

* <math>\varepsilon \in \left[\frac{1}{50}, 1\right]</math>.

The frequency response of the transfer function <math>H(s,\varepsilon) = C(sI_{N}-(\varepsilon A_{\varepsilon} + A_{0}))^{-1}B</math> is plotted for parameter values <math>\varepsilon \in \left[\frac{1}{50}, \frac{1}{20}, \frac{1}{10}, \frac{1}{5}, \frac{1}{2}, 1\right]</math> in Fig. 2.

Other interesting plots result for small values of the parameter <math>\varepsilon</math>.
For example, for <math>\varepsilon = \frac{1}{100}</math> or <math>\frac{1}{1000}</math>, the peaks in the frequency response become more pronounced, since the poles move closer to the imaginary axis.

For <math>\varepsilon \in \left[\frac{1}{50}, \frac{1}{20}, \frac{1}{10}, \frac{1}{5}, \frac{1}{2}, 1\right]</math>, we also plotted the decay of the Hankel singular values in Fig. 3.
Notice that for small values of the parameter, the decay of the Hankel singular values is very slow.

==Data and Scripts==

This benchmark includes one data set. The matrices can be downloaded in the [http://math.nist.gov/MatrixMarket/formats.html MatrixMarket] format:
* [[Media:Synth_matrices.tar.gz|Synth_matrices.tar.gz]] (1.28 kB)
The matrix name is used as an extension of the matrix file.

System data of arbitrary even order <math>N</math> can be generated in MATLAB or Octave by the following script:

<syntaxhighlight lang="octave">
N = 100; % Order of the resulting system.

% Set coefficients.
a = -linspace(1e1, 1e3, N/2).';
b = linspace(1e1, 1e3, N/2).';
c = ones(N/2, 1);
d = zeros(N/2, 1);

% Build 2x2 submatrices.
aa(1:2:N-1, 1) = a;
aa(2:2:N, 1) = a;
bb(1:2:N-1, 1) = b;
bb(2:2:N-2, 1) = 0;

% Set up system matrices.
Ae = spdiags(aa, 0, N, N);
A0 = spdiags([0; bb], 1, N, N) + spdiags(-bb, -1, N, N);
B = 2 * sparse(mod(1:N, 2)).';
C(1:2:N-1) = c.';
C(2:2:N) = d.';
C = sparse(C);
</syntaxhighlight>

or in Python:

<syntaxhighlight lang="python">
import numpy as np
import scipy.sparse as sps

N = 100 # Order of the resulting system.

# Set coefficients.
a = -np.linspace(1e1, 1e3, N//2)
b = np.linspace(1e1, 1e3, N//2)
c = np.ones(N//2)
d = np.zeros(N//2)

# Build 2x2 submatrices.
aa = np.empty(N)
aa[::2] = a
aa[1::2] = a
bb = np.zeros(N)
bb[::2] = b

# Set up system matrices.
Ae = sps.diags(aa, format='csc')
A0 = sps.diags([bb, -bb], [1, -1], (N, N), format='csc')
B = np.zeros((N, 1))
B[::2, :] = 2
C = np.empty((1, N))
C[0, ::2] = c
C[0, 1::2] = d
</syntaxhighlight>

Beside that, the plots in Fig. 1 and Fig. 2 can be generated in MATLAB and Octave using the following script:

<syntaxhighlight lang="octave">
% Get residues of the system.
r(1:2:N-1, 1) = c + 1j * d;
r(2:2:N, 1) = c - 1j * d;

ep = [1/50; 1/20; 1/10; 1/5; 1/2; 1]; % Parameter epsilon.
jw = 1j * linspace(0, 1.2e3, 5000).'; % Frequency grid.

% Computations for all given parameter values.
p = zeros(2 * length(a), length(ep));
Hjw = zeros(length(ep), 5000);
for k = 1:length(ep)
p(:, k) = [ep(k) * a + 1j * b; ep(k) * a - 1j * b]; % Poles.
[jww, pp] = meshgrid(jw, p(:, k));
Hjw(k, :) = (r.') * (1 ./ (jww - pp)); % Frequency response.
end

% Plot poles.
figure;
plot(real(p), imag(p), '.', 'MarkerSize', 12);
xlabel('Re(p)');
ylabel('Im(p)');
legend( ...
'\epsilon = 1/50', ...
'\epsilon = 1/20', ...
'\epsilon = 1/10', ...
'\epsilon = 1/5', ...
'\epsilon = 1/2', ...
'\epsilon = 1');

% Plot frequency response.
figure;
loglog(imag(jw), abs(Hjw), 'LineWidth', 2);
axis tight;
xlim([6 1200]);
xlabel('frequency (rad/sec)');
ylabel('magnitude');
legend( ...
'\epsilon = 1/50', ...
'\epsilon = 1/20', ...
'\epsilon = 1/10', ...
'\epsilon = 1/5', ...
'\epsilon = 1/2', ...
'\epsilon = 1');
</syntaxhighlight>

or in Python:

<syntaxhighlight lang="python">
import matplotlib.pyplot as plt

# Get residues of the system.
r = np.empty(N, dtype=complex)
r[::2] = c + 1j * d
r[1::2] = c - 1j * d

ep = [1/50, 1/20, 1/10, 1/5, 1/2, 1] # Parameter epsilon.
jw = 1j * np.geomspace(6, 1.2e3, 5000) # Frequency grid.

# Computations for all given parameter values.
p = np.zeros((len(ep), N), dtype=complex)
Hjw = np.zeros((len(ep), len(jw)), dtype=complex)
for k, epk in enumerate(ep):
# Poles.
p[k, :N//2] = epk * a + 1j * b
p[k, N//2:] = epk * a - 1j * b
# Frequency response.
Hjw[k, :] = (r / (jw[:, np.newaxis] - p[k])).sum(axis=1)

# Plot poles.
fig, ax = plt.subplots()
for k, epk in enumerate(ep):
ax.plot(p[k].real, p[k].imag, '.', label=fr'$\varepsilon$ = {epk}')
ax.autoscale(tight=True)
ax.set_xlabel('Re(p)')
ax.set_ylabel('Im(p)')
ax.legend()

# Plot frequency response.
fig, ax = plt.subplots()
for k, epk in enumerate(ep):
ax.loglog(jw.imag, np.abs(Hjw[k]), label=fr'$\varepsilon$ = {epk}', linewidth=2)
ax.autoscale(tight=True)
ax.set_xlabel('frequency (rad/sec)')
ax.set_ylabel('magnitude')
ax.legend()
</syntaxhighlight>

==Dimensions==

System structure:
:<math>
\begin{align}
\dot{x}(t) &= (\varepsilon A_{\varepsilon} + A_{0})x(t) + Bu(t), \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>A_{\varepsilon} \in \mathbb{R}^{N \times N}</math>,
<math>A_{0} \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>Synth_matrices</tt>: <math>N = 100</math>,
arbitrary even order <math>N</math> by using the [[#scr1|script]]

==Citation==

To cite this benchmark and its data:
::The MORwiki Community, '''Synthetic parametric model'''. hosted at MORwiki - Model Order Reduction Wiki, 2005. https://modelreduction.org/morwiki/Synthetic_parametric_model

@MISC{morwiki_synth_pmodel,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Synthetic parametric model},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/Synthetic_parametric_model}</nowiki>,
year = 2005
}

==Contact==

''[[User:Ionita]]''

Supersonic Engine Inlet

2025-06-17T05:40:13Z

Schuetz2:

[[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>{https://modelreduction.org/morwiki/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>

Stokes equation

2025-06-17T05:39:55Z

Schuetz2:

{{preliminary}} 

[[Category:benchmark]]
[[Category:procedural]]
[[Category:SISO]]
[[Category:linear]]
[[Category:Sparse]]

==Description==
This benchmark presents the two-dimensional instationary [[wikipedia:Stokes_flow|Stokes equation]],
which models flow of an incompressible fluid in a domain <ref name="Sty03"/>,<ref name="Sty04"/>,<ref name="MehS05"/>,<ref name="Sty06"/>,<ref name="Sch07"/>.
The associated partial differential equation system is given by:
:<math>
\begin{align}
\frac{\partial v}{\partial t} &= \Delta v - \nabla p + f, \qquad (x,t) \in \Omega \times (0,T] \\
0 &= \operatorname{div} v, \\
v &= 0, \qquad \qquad \qquad \quad \; (x,t) \in \partial \Omega \times (0,T]
\end{align}
</math>
with velocity variable <math>v(x,t)</math> and pressure variable <math>\rho(x,t)</math>,
on a spatial domain <math>\Omega = [0,1] \times [0,1] \subset \mathbb{R}^2</math>,
and an external forcing term <math>f</math>.
The boundary conditions are no-slip.

A finite volume discretization on a uniform, staggered grid yields the descriptor system:
:<math>
\begin{align}
\begin{bmatrix} E_{11} & 0 \\ 0 & 0 \end{bmatrix} \begin{pmatrix} \dot{v}_h(t) \\ 0 \end{pmatrix} &=
\begin{bmatrix} A_{11} & A_{12} \\ A_{12}^\intercal & 0 \end{bmatrix} \begin{pmatrix} v_h(t) \\ \rho_h(t) \end{pmatrix} +
\begin{bmatrix} B_1 \\ B_2 \end{bmatrix} u(t) \\
y(t)\quad & = \begin{bmatrix} C_1 \,\;&\;\, C_2 \end{bmatrix} \begin{pmatrix} v_h(t) \\ \rho_h(t) \end{pmatrix}
\end{align}
</math>
The matrix <math>A_{11}</math> matrix is the discretized Laplace operator,
while <math>A_{12}</math> corresponds to the discrete gradient and divergence operators.
For this benchmark the compound discretization of the boundary values and external forcing <math>[B_1 \; B_2]^\intercal \in \mathbb{R}^{N \times 1}</math> is chosen (uniformly) randomly,
whereas the output matrix <math>[C_1 \; C_2] \in \mathbb{R}^{1 \times N}</math> is set to:
:<math>
\begin{align}
\begin{bmatrix} C_1 & C_2 \end{bmatrix} = \begin{bmatrix} 1 & 0 & \dots & 0 \end{bmatrix}.
\end{align}
</math>

==Data==

This is a procedural benchmark.
A MATLAB m-file to generate <math>E, A, B, C</math> matrices can be found as part of the [https://www.mpi-magdeburg.mpg.de/projects/mess M.E.S.S] project,
under:

DEMOS/models/stokes/stokes_ind2.m

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

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Stokes equation'''. MORwiki - Model Order Reduction Wiki, 2018. https://modelreduction.org/morwiki/Stokes_equation

@MISC{morwiki_stokes,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Stokes equation},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/Stokes_equation}</nowiki>,
year = {20XX}
}

* For the background on the benchmark:

@PHDTHESIS{Sch07,
author = <nowiki>{M.Schmidt}</nowiki>,
title = {Systematic discretization of input/output maps and other contributions to the control of distributed parameter systems},
school = {TU Berlin},
year = {2007},
doi = {10.14279/depositonce-1600}
}

==References==

<references>

<ref name="Sty03">T. Stykel. [https://doi.org/10.1002/pamm.200310302 Balanced truncation model reduction for descriptor systems], Proceedings in Applied Mathematics and Mechanics 3: 5--8, 2003.</ref>

<ref name="Sty04">T. Stykel. [https://doi.org/10.1007/s00498-004-0141-4 Gramian-Based Model Reduction for Descriptor System], Mathematics of Control, Signals, and Systems 16(4): 297--319, 2004.</ref>

<ref name="MehS05">V. Mehrmann, T. Stykel. [https://doi.org/10.1007/3-540-27909-1_3 Balanced Truncation Model Reduction for Large-Scale Systems in Descriptor Form], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 83--115, 2005.</ref>

<ref name="Sty06">T. Stykel. [https://doi.org/10.1016/j.laa.2004.01.015 Balanced Truncation model reduction for semidiscretized Stokes equation], Linear Algebra and its Application 415(2--3): 262--289, 2006.</ref>

<ref name="Sch07">M.Schmidt. [https://doi.org/10.14279/depositonce-1600 Systematic discretization of input/output maps and other contributions to the control of distributed parameter systems], Ph.D. thesis, TU Berlin, 2007.</ref>

</references>

==Contact==

[[User:Himpe]]

Scanning Electrochemical Microscopy

2025-06-17T05:39:17Z

Schuetz2:

[[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. https://modelreduction.org/morwiki/Scanning_Electrochemical_Microscopy

@MISC{morwiki_secm,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Scanning Electrochemical Microscopy},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/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

2025-06-17T05:38:32Z

Schuetz2:

[[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>{https://modelreduction.org/morwiki/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>

Randomly Generated

2025-06-17T05:38:05Z

Schuetz2:

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

==Description==

This '''randomly generated''' state-space system is a procedural SISO test system for model reduction of linear time-invariant systems from <ref name="willcox02"/>.
All matrices are generated from a uniformly random distribution: the system matrix is a diagonal matrix with elements in <math>[-1,0]</math>, where as the input and output vectors are drawn from <math>[0,1]</math>.
The generated systems are stable and minimal, due to structure of <math>N</math> disconnected subsystems.



==Data==

The following Matlab code assembles the above described <math>A</math>, <math>B</math> and <math>C</math> matrix for a given state-space dimension <math>N</math> and optionally a seed <math>S</math> for the random number generator.

<div class="thumbinner" style="width:540px;text-align:left;">
<source lang="matlab">
function [A,B,C] = rnd(N,S)

if(nargin>1 && not(isempty(S))), rand('seed',S); end;

A = spdiags(-rand(N,1),0,N,N);
B = rand(N,1);
C = rand(1,N);
end
</source>
</div>

==Dimensions==

System structure:
:<math>
\begin{align}
\dot{x}(t) &= Ax(t) + Bu(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 1}</math>,
<math>C \in \mathbb{R}^{1 \times N}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_rnd,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Randomly Generated},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/Randomly_Generated}</nowiki>,
year = {2018}
}

* For the background on the benchmark:

@ARTICLE{morWilP02,
author = <nowiki>{K. Willcox and J. Peraire}</nowiki>,
title = {Balanced Model Reduction via the Proper Orthogonal Decomposition},
journal = {AIAA Journal},
volume = {40},
number = {11},
pages = {2323--2330},
year = {2002},
doi = {10.2514/2.1570}
}

==Reference==

<references>

<ref name="willcox02">K. Willcox and J. Peraire. "[https://doi.org/10.2514/2.1570 Balanced Model Reduction via the Proper Orthogonal Decomposition]", AIAA Journal, 40(11): 2323--2330, 2002.</ref>

</references>

==Contact==

''[[User:Himpe|Christian Himpe]]''

Peek Inductor

2025-06-17T05:37:13Z

Schuetz2:

[[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>{https://modelreduction.org/morwiki/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>

Nonlinear RC Ladder

2025-06-17T05:36:43Z

Schuetz2:

[[Category:benchmark]]
[[Category:procedural]]
[[Category:nonlinear]]
[[Category:SISO]]

==Description==
<figure id="fig:nrcl">[[File:nrcl.png|400px|thumb|right| Nonlinear RC-Ladder]]</figure>

The nonlinear RC-ladder is an electronic test circuit first introduced in <ref name="chen99"/>, and its variant is also introduced in <ref name="RewW03"/>.
These nonlinear first-order system model a resistor-capacitor network that exhibits a distinct nonlinear behaviour caused by either the nonlinear resistors consisting of a parallel connected resistor with a diode (see the right figure) or the nonlinear resistors connected parallel to the capacitor (see Fig. 7 in <ref name = "RewW03"/>).

===Model 1===
First, we discuss the modelling of an RC circuit, where the nonlinear resistors consist of a parallel connected resistor with a diode as shown in the above figure. For this, the underlying model is given by a SISO gradient system of the form <ref name="condon04"/>:

:<math>
\dot{x}(t) = \begin{pmatrix} -g(x_1(t)) - g(x_1(t) - x_2(t)) \\ g(x_1(t)-x_2(t)) - g(x_2(t)-x_3(t)) \\ \vdots \\ g(x_{k-1}(t) - x_k(t)) - g(x_k(t) - x_{x+1}(t)) \\ \vdots \\ g(x_{N-1}(t) - x_N(t)) \end{pmatrix}+\begin{pmatrix}u(t) \\ 0 \\ \vdots \\ 0 \\ \vdots \\ 0 \end{pmatrix},
</math>

:<math>
y(t) = x_1(t),
</math>

where the <math>g</math> is a mapping <math>g(x_i):\mathbb{R} \to \mathbb{R}</math>:

:<math>
g(x_i) = g_D(x_i) + x_i,
</math>

which combines the effect of a [[wikipedia:Diode|diode]] and a resistor.

====Nonlinearity====
The nonlinearity <math>g_D</math> models a diode as a nonlinear resistor,
based on the [[wikipedia:Diode_modelling#Shockley_diode_model|Shockley model]] <ref name="reis14"/>:

:<math>
g_D(x_i) = i_S (\exp(u_P x_i) - 1),
</math>

with material parameters <math>i_S > 0</math> and <math>u_P > 0</math>.

For this benchmark the parameters are selected as: <math>i_S = 1</math> and <math>u_P = 40</math> as in <ref name="chen99"/>.

===Model 2===
Second, we discuss the modelling of an RC circuit, where the nonlinear resistors are connected parallel to the capacitors (see Fig. 7 in <ref name = "RewW03"/>). For this, the underlying model is also given by a SISO gradient system of the form <ref name="RewW03"/>:

:<math>
\dot{x}(t) = \begin{pmatrix} -2 & 1& & \\ 1 & -2 & 1 & \\ & \ddots & \ddots & \ddots\\ & & 1 & -2 \end{pmatrix} x(t) + g(x),
</math>

and the output function:

:<math>
y(t) = x_1(t),
</math>

where the <math>g</math> is a mapping <math>g(x_i):\mathbb{R} \to \mathbb{R}</math>, respresenting the effect of a nonlinear resistor.

:<math>
g(x_i) = a\cdot \text{sgn}(x_i)\cdot x_i^2
</math>

which represents the effect of a nonlinear resistor, and sgn denotes the [[wikipedia:Sign_function|sign function]].

====Nonlinearity====
The nonlinearity <math>g_D</math> models a diode as a nonlinear resistor <ref name="RewW03"/>:

:<math>
g(x_i) = a\cdot \text{sgn}(x_i)\cdot x_i^2
</math>

which represents the effect of a nonlinear resistor, and sgn denotes the [[wikipedia:Sign_function|sign function]].

For this benchmark the parameters are selected as: <math>a = 1</math> as in <ref name="chen99"/>.

Alternatively, the variant from <ref name="kawano19"/> can be used, featuring the nonlinearity:

:<math>
g(x_i) = \frac{x_i^2}{2} + \frac{x_i^3}{3}.
</math>

This represents practically a resistor-inductor cascade with nonlinear resistors.

===Model 3===
Third, a circuit of chained [[wikipedia:Inverter_(logic_gate)|inverter gates]], a so-called inverter chain <ref name="gu12"/>, is presented.
This is a SISO system, given by:

:<math>
\dot{x}(t) = -x(t) + \begin{pmatrix} 0 \\ g(x_1(t)) \\ \vdots \\ g(x_{N-1}(t)) \end{pmatrix} + \begin{pmatrix} u(t) \\ 0 \\ \vdots \\ 0 \end{pmatrix},
</math>

and the output function:

:<math>
y(t) = x_N(t).
</math>

====Nonlinearity====

The nonlinear function <math>g</math> is a mapping <math>g(x_i):\mathbb{R} \to \mathbb{R}</math> describing the inverter characteristic:

:<math>
g(x_i) = v \tanh(a x_i),
</math>

being a parameterized [[wikipedia:Hyperbolic_function|hyperbolic tangent]] with the supply voltage <math>v=1</math> and a physical parameter <math>a</math>, i.e. <math>a=5</math>.

===Input===

As an external input, several alternatives are presented in <ref name="chen00"/>, which are listed next.
A simple step function is given by:
:<math>
u_1(t)=\begin{cases}0 & t < 4 \\ 1 & t \geq 4 \end{cases},
</math>

an exponential decaying input is provided by:
:<math>
u_2(t) = e^{-t}.
</math>

Additional input sources are given by conjunction of sine waves with different periods <ref name="condon04a"/>:
:<math>
u_3(t) = \sin(2\pi 50t)+\sin(2\pi 1000t),
</math>

:<math>
u_4(t) = \sin(2\pi 50t) \sin(2\pi 1000t).
</math>

==Data==

A sample procedural MATLAB implementation for order <math>N</math> for all three models is given by:

<div class="thumbinner" style="width:540px;text-align:left;">
<source lang="matlab">

function [f,B,C] = nrc(N,model)
%% Procedural generation of "Nonlinear RC Ladder" benchmark system

B = sparse(1,1,1,N,1); % input matrix
C = sparse(1,1,1,1,N); % output matrix

switch lower(model)

case 'shockley'

g = @(x) exp(40.0*x) + x - 1.0;

A0 = sparse(1,1,1,N,N);

A1 = spdiags(ones(N-1,1),-1,N,N) - speye(N);
A1(1,1) = 0;

A2 = spdiags([ones(N-1,1);0],0,N,N) - spdiags(ones(N,1),1,N,N);

f = @(x) -g(A0*x) + g(A1*x) - g(A2*x);

case 'sign'

A = gallery('tridiag',N,1,-2,1);

f = @(x) A*x - sign(x).*(x.*x);

case 'ind'

A = gallery('tridiag',N,1,-2,1);

f = @(x) A*x - ((x.^2)./2 + (x.^3)./3);

case 'inv'

f = @(x) [0;tanh(x(2:end))] - x;

C = sparse(1,N,1,1,N);

otherwise

error('Choose shockley, sign, ind or inv');
end
end

</source>
</div>

Here the vector field is realized in a vectorized form as a [[wikipedia:Closure_(computer_programming)|closure]].

==Dimensions==

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

System dimensions:

<math>f : \mathbb{R}^N \to \mathbb{R}^N</math>,
<math>B \in \mathbb{R}^{N \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times N}</math>.

==Citation==

To cite these benchmarks, use the following references:

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

@MISC{morwiki_modNonRCL,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Nonlinear RC Ladder},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/Nonlinear_RC_Ladder}</nowiki>,
year = 2018
}

* For the background on the benchmarks:
** for Model 1 [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morChe99 morChe99] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morChe99 BibTeX])
** for Model 2 [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morRew03 morRew03] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morRew03 BibTeX])
** for Model 3 [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morGu12 morGu12] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morGu12 BibTeX])

==References==

<references>

<ref name="chen99">Y. Chen, "[http://hdl.handle.net/1721.1/9381 Model Reduction for Nonlinear Systems]", Master Thesis, 1999.</ref>

<ref name="RewW03">M. Rewienski and J. White, "[https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1174092 A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices]", IEEE Transactions on computer-aided design of integrated circuits and systems 22(2): 155--170, 2003.</ref>

<ref name="chen00">Y. Chen and J. White, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.19.8951&rep=rep1&type=pdf A Quadratic Method for Nonlinear Model Order Reduction]", Int. Conference on Modelling and Simulation of Microsystems Semiconductors, Sensors and Actuators, 2000.</ref>

<ref name="condon04">M. Condon and R. Ivanov, "[https://doi.org/10.1007/s00332-004-0617-5 Empirical Balanced Truncation for Nonlinear Systems]", Journal of Nonlinear Science 14(5):405--414, 2004.</ref>

<ref name="condon04a">M. Condon and R. Ivanov, "[https://doi.org/10.1108/03321640410510730 Model Reduction of Nonlinear Systems]", COMPEL 23(2): 547--557, 2004</ref>

<ref name="gu12">C. Gu, "[https://www2.eecs.berkeley.edu/Pubs/TechRpts/2012/EECS-2012-217.pdf Model Order Reduction of Nonlinear Dynamical Systems]", PhD Thesis (University of California, Berkeley), 2012.</ref>

<ref name="reis14">T. Reis. "[https://doi.org/10.1007/978-3-319-08437-4_2 Mathematical Modeling and Analysis of Nonlinear Time-Invariant RLC Circuits]", In: Large-Scale Networks in Engineering and Life Sciences. Modeling and Simulation in Science, Engineering and Technology: 125--198, 2014.</ref>

<ref name="kawano19">Y. Kawano, J.M.A. Scherpen. "[https://arxiv.org/abs/1902.09836 Empirical Differential Gramians for Nonlinear Model Reduction]", arXiv (cs.SY): 1902.09836, 2019.</ref>

</references>

==Contact==

''[[User:Himpe|Christian Himpe]]''

Nonlinear Heat Transfer

2025-06-17T05:36:23Z

Schuetz2:

[[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. https://modelreduction.org/morwiki/Nonlinear_Heat_Transfer

@MISC{morwiki_nheat,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Nonlinear Heat Transfer},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/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>

Modified Gyroscope

2025-06-17T05:35:21Z

Schuetz2:

[[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. https://modelreduction.org/morwiki/Modified_Gyroscope

@MISC{morwiki_modgyro,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Modified Gyroscope},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/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]] ''

Micropyros Thruster

2025-06-17T05:34:41Z

Schuetz2:

[[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>{https://modelreduction.org/morwiki/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>

Mass-Spring-Damper

2025-06-17T05:34:19Z

Schuetz2:

{{preliminary}} 

[[Category:benchmark]]
[[Category:nonlinear]]
[[Category:ODE]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:second differential order]]
[[Category:Procedural]]
[[Category:SISO]]

==Description: Mass-Spring-Damper System==

This benchmark is a generalization of the nonlinear [[wikipedia:Mass-spring-damper_model|mass-spring-damper]] system presented in <ref name="kawano15"/>,
which is concerned with modeling the a mechanical systems consisting of chained masses, linear and nonlinear springs, and dampers.
The underlying mathematical model is a second order system:
:<math>
\begin{align}
M \ddot{x}(t) + D \dot{x}(t) + K x(t) + f(x(t)) &= B u(t), \\
y(t) &= C x(t).
\end{align}
</math>

===First Order Representation===

The second order system can be represented as a first order system as follows:

:<math>
\begin{align}
\begin{pmatrix} 1 & 0 \\ 0 & M \end{pmatrix} \begin{pmatrix} \dot{p} \\ \dot{v} \end{pmatrix} &=
\begin{pmatrix} 0 & 1 \\ K & D \end{pmatrix} \begin{pmatrix} p \\ v \end{pmatrix} +
\begin{pmatrix} 0 \\ f_p(p) \end{pmatrix} + \begin{pmatrix} 0 \\ B_v \end{pmatrix} \\
y &= \begin{pmatrix} C_p & 0 \end{pmatrix} \begin{pmatrix} p \\ v \end{pmatrix}
\end{align}
</math>

with the components:

:<math>
M = m \begin{pmatrix} 1 \\ & \ddots \end{pmatrix}, \quad
K_0 = k_l \begin{pmatrix} -2 & 1 \\ 1 & -2 & \ddots \\ & \ddots & \ddots \end{pmatrix}, \quad
D = d \begin{pmatrix} -2 & 1 \\ 1 & -2 & \ddots \\ & \ddots & \ddots \end{pmatrix}, \quad
B_v = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 \end{pmatrix}, \quad
C_p = \begin{pmatrix} 0 & \dots & 0 & 1 \end{pmatrix},
</math>

and the nonlinear term:

:<math>
f_p(p) = -k_n \Big( \begin{pmatrix} 1 & -1 \\ & \ddots & \ddots \end{pmatrix} p \Big)^3 -k_n \Big( \begin{pmatrix} 1 \\ -1 & \ddots \\ & \ddots \end{pmatrix} p \Big)^3
</math>

and thus yielding the classic first order components:

:<math>
E = \begin{pmatrix} 1 & 0 \\ 0 & M \end{pmatrix}, \quad
A = \begin{pmatrix} 0 & 1 \\ K & D \end{pmatrix}, \quad
B = \begin{pmatrix} 0 \\ B_v \end{pmatrix}, \quad
C = \begin{pmatrix} C_p & 0 \end{pmatrix}.
</math>

The parameters for the mass <math>m</math>, linear spring constant <math>k_l</math>, nonlinear spring constant <math>k_n</math>, and damping <math>d</math> are chosen in <ref name="kawano15"/> as <math>m=1</math>, <math>k_l=1</math>, <math>k_n=2</math>, and <math>d=1</math>.

==Data==

The following Matlab code assembles the above described <math>E</math>, <math>A</math>, <math>B</math> and <math>C</math> parameter dependent matrices and the function <math>f</math> for a given number of subsystems <math>N</math>.

<div class="thumbinner" style="width:540px;text-align:left;">
<source lang="matlab">
function [E,A,B,C,f] = msd(N)

U = speye(N); % Sparse unit matrix
T = gallery('tridiag',N,-1,2,-1); % Sparse tridiagonal matrix
H = gallery('tridiag',N,-1,1,0); % Sparse transport matrix
Z = sparse(N,N); % Sparse all zero matrix
z = sparse(N,1); % Sparse all zero vector

E = @(m) [U,Z;Z,m*U]; % Handle to parametric E matrix
A = @(kl,d) [Z,U;kl*T,d*T]; % Handle to parametric A matrix
B = sparse(2*N,1,1,2*N,1);
C = sparse(N,1,1,2*N,1);
f = @(x,kn) [z;-kn*( (H'*x(N+1:end)).^3 - (H*x(N+1:end)).^3)];
end
</source>
</div>

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
E(m) \dot{x}(t) &=& A(k_l,d)x(t) + f(x(t);k_n) + Bu(t) \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

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

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_msd,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Mass-Spring-Damper System},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/Mass-Spring-Damper}</nowiki>,
year = 2018
}

* For the background on the benchmark:

@INPROCEEDINGS{morKawS15,
title = {Model Reduction by Generalized Differential Balancing},
author = {Y. Kawano and J.M.A. Scherpen},
booktitle = {Mathematical Control Theory I: Nonlinear and Hybrid Control Systems},
series = {Lecture Notes in Control and Information Sciences},
volume = {461},
pages = {349--362},
year = {2015},
doi = {10.1007/978-3-319-20988-3}
}

==References==

<references>

<ref name="kawano15">Y. Kawano and J.M.A. Scherpen, [https://doi.org/10.1007/978-3-319-20988-3_19 Model Reduction by Generalized Differential Balancing], In: Mathematical Control Theory I: Nonlinear and Hybrid Control Systems, Lecture Notes in Control and Information Sciences 461: 349--362, 2015.</ref>

</references>

Low-Pass Butterworth Filter

2025-06-17T05:33:26Z

Schuetz2:

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

==Description==

This benchmark from <ref name="antoulas01"/> represents an analogue '''low-pass Butterworth filter'''.
A [[wikipedia:Butterworth_filter|Butterworth filter]] is an electrical signal processing filter.
The [[wikipedia:Low-pass_filter|low-pass]] variant allows signals with frequencies below a cut-off frequency <math>f</math> to pass and mitigates those with frequencies above <math>f</math>.

==Data==

The following Matlab code assembles the above described <math>A</math>, <math>B</math> and <math>C</math> matrix for a given state-space dimension <math>N</math> and a cut-off frequency <math>f=1 rad/s</math>.

<div class="thumbinner" style="width:540px;text-align:left;">
<source lang="matlab">
[A,B,C,D] = butter(N,1,'low','s'); % D = 0
</source>
</div>

The <tt>butter</tt> method is part of the [https://www.mathworks.com/products/control.html Control System Toolbox]; an example for <math>N=100</math> is provided here: [[File:Lpbw_mat.zip|Lpbw_mat.zip]]

==Dimensions==

System structure:
:<math>
\begin{align}
\dot{x}(t) &= Ax(t) + Bu(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 1}</math>,
<math>C \in \mathbb{R}^{1 \times N}</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Low-Pass Butterworth Filter'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Low-Pass_Butterworth_Filter

@MISC{morwiki_lpbw,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Low-Pass Butterworth Filter},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/Low-Pass_Butterworth_Filter}</nowiki>,
year = {2018}
}

* For the background on the benchmark:

@ARTICLE{morAntSS01,
author = <nowiki>{A.C. Antoulas and 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},
url = <nowiki>{http://www.ams.org/books/conm/280/4630}</nowiki>
}

==Reference==

<references>

<ref name="antoulas01">A.C. Antoulas, D.C. Sorenson and S. Gugercin. "[http://www.ams.org/books/conm/280/4630 A Survey of Model Reduction Methods for Large-Scale Systems]", Contemporary Mathematics, 280: 193--219, 2001.</ref>

</references>

==Contact==

''[[User:Himpe|Christian Himpe]]''

Linear Advection

2025-06-17T05:33:10Z

Schuetz2:

[[Category:benchmark]]
[[Category:procedural]]
[[Category:SISO]]
[[Category:Sparse]]

==Description==

The '''Linear Advection''' benchmark is an input-output system based on the one-dimensional [[wikipedia:Advection#The_advection_equation|linear advection equation]] (also known as linear transport equation):
:<math>
\begin{align}
\partial_t z = -a \, \partial_x z,
\end{align}
</math>
which is a first order hyperbolic partial differential equation.

For this benchmark, the left-hand-side boundary of the domain is considered to be the input <math>u(t)</math>,
:<math>
\begin{align}
z(0,t) = u(t),
\end{align}
</math>
while the right-hand-side boundary is selected as output <math>y(t)</math>:
:<math>
\begin{align}
y(t) = z(1,t),
\end{align}
</math>
whereas the domain is assumed to be of unit length and <math>a > 0</math> describes the transport velocity.

A spatial discretization, here with a simple [[wikipedia:Upwind_differencing_scheme_for_convection|upwind finite difference scheme]] leads to the standard input-output system form:
:<math>
\begin{align}
\dot{x}(t) &= Ax(t) + Bu(t), \\
y(t) &= Cx(t),
\end{align}
</math>
with matrices:
:<math>
\begin{align}
A = -\frac{a}{\Delta x} \begin{pmatrix} -1 & & & 0 \\ 1 & -1 \\ & \ddots & \ddots \\ 0 & & 1 & -1 \end{pmatrix}, \quad
B = \frac{1}{\Delta x} \begin{pmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \quad
C = \begin{pmatrix} 0 & \dots & 0 & 1 \end{pmatrix}.
\end{align}
</math>
The challenge of this SISO system benchmark lies in its hyperbolicity.
Even though this system seems simple,
its numerical simulation and especially order reduction can be complex due to the transport phenomenon<ref name="zulfiqar18"/>,
which is known in control theoretic terms as [https://de.wikipedia.org/Totzeit_(Regelungstechnik) dead time].
Projection-based model reduction is generally not well suited for systems derived from hyperbolic PDEs<ref name="breiten21"/>,
especially, energy-based method tend to produce unstable reduced order models<ref name="benner18"/> due to the non-normal system matrix.
This system can be used to model flame-acoustic interaction<ref name="meindl16"/>.

==Data==
To procedurally instantiate the state-space components for this benchmark model,
it is assumed the reciprocal spatial resolution <math>N := \frac{1}{\Delta x} \in \mathbb{N}</math> is given,
which determines the order of the system, alongside the transport velocity <math>a</math>.

<div class="thumbinner" style="width:540px;text-align:left;">
<source lang="matlab">
function [A,B,C] = lte(a,N)

A = -(abs(a) * N) * spdiags(ones(N,1) * [-1,1],[-1,0],N,N);
B = N * sparse(1,1,1,N,1);
C = sparse(1,N,1,1,N);
end
</source>
</div>

==Dimensions==

System structure:

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

==Citation==

To cite this benchmark and its data:
::The MORwiki Community, '''Linear Advection'''. hosted at MORwiki - Model Order Reduction Wiki, 2019. http://modelreduction.org/index.php/Linear_Advection

@MISC{morwiki_linad,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Linear Advection},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/Linear_Advection}</nowiki>,
year = 2019
}

==References==

<references>

<ref name="meindl16">M. Meindl, T. Emmert, W. Polifke. "[https://www.researchgate.net/profile/Wolfgang-Polifke/publication/322077228_Efficient_calculation_of_thermoacoustic_modes_utilizing_state-space_models/links/5a43465ea6fdcce1971653ee/Efficient-calculation-of-thermoacoustic-modes-utilizing-state-space-models.pdf Efficient Calculation of Thermoacoustic Modes Utilizing State-Space Models]". Proceedings of the 23rd International Congress on Sound & Vibration: 1--8, 2016.</ref>

<ref name="benner18">P. Benner, C. Himpe, T. Mitchell. "[https://doi.org/10.1007/s10444-018-9592-x On Reduced Input-Output Dynamic Mode Decomposition]". Advances in Computational Mathematics (ModRed Special Issue) 44(6): 1751--1768, 2018.</ref>

<ref name="zulfiqar18">U. Zulfiqar, M. Liaquat. "[https://doi.org/10.3906/elk-1706-124 Model reduction of discrete-time systems in limited intervals]". Turkish Journal of Electrical Engineering and Computer Sciences 26: 294--306, 2018.</ref>

<ref name="breiten21">T. Breiten, T. Stykel. "[https://doi.org/10.1515/9783110498967-002 Balancing-related model reduction methods]". Volume 1 System- and Data-Driven Methods and Algorithms: 15--56, 2021.</ref>

</references>

==Contact==

''[[User:Himpe|Christian Himpe]]''

Linear 1D Beam

2025-06-17T05:32:52Z

Schuetz2:

[[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>{https://modelreduction.org/morwiki/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>

Inverse Lyapunov Procedure

2025-06-17T05:32:31Z

Schuetz2:

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

==Description==
The '''Inverse Lyapunov Procedure''' (ILP) is a synthetic random linear system generator.
It is based on reversing the [[Balanced_Truncation|Balanced Truncation]] procedure and was developed in <ref name="smith03"/>,
where a description of the algorithm is given.
In aggregate form, for randomly generated controllability and observability gramians, a balancing transformation is computed.
The balanced gramian is the basis for an associated state-space system,
which is determined by solving a [[wikipedia:Lyapunov_equation|Lyapunov equation]] and then unbalanced.
A central point is the solution of the Lyapunov equations for the system matrix instead of the gramian matrix.
This is feasable due to the symmetric (semi-)positive definiteness of the gramians and the requirement for a stable system, yet with a non-unique solution.
Following, the steps of the ILP are listed:

# Sample eigenvalues of controllability and observability Gramians.
# Generate random orthogonal matrices (ie.: SVD of random matrix).
# Compute balancing transformation for these Gramians.
# Sample random input and output matrices.
# Scale output matrix to input matrix.
# Solve Lyapunov equation for system matrix.
# Unbalance system.

===Inverse Sylvester Procedure===
A variant of the '''Inverse Lyapunov Procedure''' is the inverse Sylvester procedure (ILS) <ref name="himpe17"/>,
which generates only state-space symmetric systems.
Instead of balanced truncation, the [[wikipedia:Cross_Gramian|cross Gramian]] is utilized for the random system generation, and hence a [[wikipedia:Sylvester_equation|Sylvester equation]] is needs to be solved.
The steps for the ILS are listed below:

# Sample cross Gramian eigenvalues.
# Sample random input matrix, and set output matrix as its transpose.
# Solve Sylvester equation for system matrix.
# Sample orthogonal unbalancing transformation (QR of random matrix).
# Unbalance system.

Even though the ILS is more limited than the ILP, for large systems it can be more efficient.

==Data==
This benchmark is procedural and the input, state and output dimensions can be chosen.
Use the following [http://matlab.com MATLAB] code to generate a random system as described above:

<div class="thumbinner" style="width:20%;text-align:left;">
<source lang="matlab">

function [A,B,C] = ilp(M,N,Q,s,r)
% ilp (inverse lyapunov procedure)
% by Christian Himpe, 2013--2018
% released under BSD 2-Clause License
%*
if(nargin==5)
rand('seed',r);
randn('seed',r);
end;

% Gramian Eigenvalues
WC = exp(0.5*rand(N,1));
WO = exp(0.5*rand(N,1));

% Gramian Eigenvectors
[P,S,R] = svd(randn(N));

% Balancing Transformation
WC = P*diag(sqrt(WC))*P';
WO = R*diag(sqrt(WO))*R';
[U,D,V] = svd(WC*WO);

% Input and Output
B = randn(N,M);

if(nargin>=4 && s~=0)
C = B';
else
C = randn(Q,N);
end

% Scale Output Matrix
BB = sum(B.*B,2); % = diag(B*B')
CC = sum(C.*C,1)'; % = diag(C'*C)
C = bsxfun(@times,C,sqrt(BB./CC)');

% Solve System Matrix
A = -sylvester(D,D,B*B');

% Unbalance System
A = V*A*U';
B = V*B;
C = C*U';

end
</source>
</div>

The function call requires three parameters; the number of inputs <math>M</math>, of states <math>N</math> and outputs <math>Q</math>.
Optionally, a symmetric system can be enforced with the parameter <math>s \neq 0</math>.
For reproducibility, the random number generator seed can be controlled by the parameter <math>r \in \mathbb{N}</math>.
The return value consists of three matrices; the system matrix <math>A</math>, the input matrix <math>B</math> and the output matrix <math>C</math>.

:<source lang="matlab">
[A,B,C] = ilp(M,N,Q,s,r);
</source>

A variant of the above code using empirical Gramians instead of a matrix equation solution can be found at http://gramian.de/utils/ilp.m , which may yield preferable results.

==Dimensions==

System structure:
:<math>
\begin{align}
\dot{x}(t) &= Ax(t) + Bu(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 M}</math>,
<math>C \in \mathbb{R}^{Q \times N}</math>.

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

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

@MISC{morwiki-invlyapproc,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Inverse Lyapunov Procedure},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/Inverse_Lyapunov_Procedure}</nowiki>,
year = 2018
}

* For the background on the benchmark:

@INPROCEEDINGS{SmiF03,
author = {Smith, S.~C. and Fisher, J.},
title = {On generating random systems: a gramian approach},
booktitle = {Proc. Am. Control. Conf.},
volume = 3,
pages = {2743--2748},
year = 2003,
doi = {10.1109/ACC.2003.1243494}
}

==References==
<references>

<ref name="smith03">S.C. Smith, J. Fisher, "[https://doi.org/10.1109/ACC.2003.1243494 On generating random systems: a gramian approach]", Proceedings of the American Control Conference, 3: 2743--2748, 2003.</ref>

<ref name="himpe17">C. Himpe, M. Ohlberger, "[https://doi.org/10.1007/978-3-319-58786-8_17 Cross-Gramian-Based Model Reduction: A Comparison]", In: Model Reduction of Parametrized Systems, Modeling, Simulation and Applications, vol. 17: 271--283, 2017.</ref>

</references>

==Contact==
[[User:Himpe|Christian Himpe]]

Gas Sensor

2025-06-17T05:31:52Z

Schuetz2:

[[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>{https://modelreduction.org/morwiki/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]

Flexible Space Structures

2025-06-17T05:31:19Z

Schuetz2:

[[Category:benchmark]]
[[Category:linear]]
[[Category:second differential order]]
[[Category:Procedural]]
[[Category:SISO]]

==Description==

The '''flexible space structure''' benchmark <ref name="gawronski90"/>,<ref name="gawronski91"/>,<ref name="gawronski96"/> is a procedural modal model which represents structural dynamics with a selectable number actuators and sensors. This model is used for truss structures in space environments i.e. the COFS-1 (Control of Flexible Structures) mast flight experiment <ref name="horner86"/>,<ref name="horta86"/>.

===Model===

In modal form the '''flexible space structure''' model for <math>K</math> modes, <math>M</math> actuators and <math>Q</math> sensors is of second order and given by:

:<math>\ddot{\nu}(t) = (2 \xi \circ \omega) \circ \dot{\nu}(t) + (\omega \circ \omega) \circ \nu = Bu(t)</math>

:<math>y(t) = C_r\dot{\nu}(t) + C_d\nu(t)</math>

with the parameters <math>\xi \in \mathbb{R}_{>0}^K</math> (damping ratio), <math>\omega \in \mathbb{R}_{>0}^K</math> (natural frequency) and using the Hadamard product <math>\circ</math>.
The first order representation follows for <math>x(t) = (\dot{\nu}(t), \omega_1\nu_1, \dots, \omega_K\nu_K)</math> by:

:<math>\dot{x}(t) = Ax(t) + Bu(t) </math>

:<math>y(t) = Cx(t)</math>

with the matrices:

:<math>A := \begin{pmatrix} A_1 & & \\ & \ddots & \\ & & A_K \end{pmatrix}, \; B := \begin{pmatrix} B_1 \\ \vdots \\ B_K \end{pmatrix}, \; C := \begin{pmatrix} C_1 & \dots & C_K \end{pmatrix}, </math>

and their components:

:<math>A_k := \begin{pmatrix} -2\xi_k\omega_k & -\omega_k \\ \omega_k & 0 \end{pmatrix}, \; B_k := \begin{pmatrix} b_k \\ 0 \end{pmatrix}, \; C_k := \begin{pmatrix} c_{rk} & \frac{c_{dk}}{\omega_k} \end{pmatrix},</math>

where <math>b_k \in \mathbb{R}^{1 \times M}</math> and <math>c_{rk}, c_{dk} \in \mathbb{R}^{Q \times 1}</math>.

===Benchmark Specifics===

For this benchmark the system matrix is block diagonal and thus chosen to be sparse.
The parameters <math>\xi</math> and <math>\omega</math> are sampled from a uniform random distributions <math>\mathcal{U}_{[0,\frac{1}{1000}]}^K</math> and <math>\mathcal{U}_{[0,100]}^K</math> respectively.
The components of the input matrix <math>b_k</math> are sampled form a uniform random distribution <math>\mathcal{U}_{[0,1]}</math>,
while the output matrix <math>C</math> is sampled from a uniform random distribution <math>\mathcal{U}_{[0,10]}</math> completely w.l.o.g, since if the components of <math>C_d</math> are random their scaling can be ignored.

==Data==

The following Matlab code assembles the above described <math>A</math>, <math>B</math> and <math>C</math> matrix for a given number of modes <math>K</math>, actuators (inputs) <math>M</math> and sensors (outputs) <math>Q</math>.

<div class="thumbinner" style="width:540px;text-align:left;">
<source lang="matlab">
function [A,B,C] = fss(K,M,Q)

rand('seed',1009);
xi = rand(1,K)*0.001; % Sample damping ratio
omega = rand(1,K)*100.0; % Sample natural frequencies

A_k = cellfun(@(p) sparse([-2.0*p(1)*p(2),-p(2);p(2),0]), ...
num2cell([xi;omega],1),'UniformOutput',0);

A = blkdiag(A_k{:});

B = kron(rand(K,M),[1;0]);

C = 10.0*rand(Q,2*K);
end
</source>
</div>

==Dimensions==

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

System dimensions:

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

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki-flexspacstruc,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Flexible Space Structures},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/Flexible_Space_Structures}</nowiki>,
year = 2018
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morGawW91 morGawW91] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morGawW91 BibTeX])

==Reference==

<references>

<ref name="gawronski90">W. Gawronski and J.N. Juang. "[https://doi.org/10.1016/B978-0-12-012736-8.50010-3 Model Reduction for Flexible Structures]", Control and Dynamic Systems, 36: 143--222, 1990.</ref>

<ref name="gawronski91">W. Gawronski and T. Williams, "[http://doi.org/10.2514/3.20606 Model Reduction for Flexible Space Structures]", Journal of Guidance 14(1): 68--76, 1991</ref>

<ref name="gawronski96">W. Gawronski. "[https://doi.org/10.1007/3540760172_4 Model reduction]". In: Balanced Control of Flexible Structures. Lecture Notes in Control and Information Sciences, vol 211: 45--106, 1996.</ref>

<ref name="horner86">G.C. Horner. "[https://ntrs.nasa.gov/search.jsp?R=19870006596 COFS-1 Research Overview]". NASA / DOD Control Structures Interaction Technology: 233--251, 1986</ref>

<ref name="horta86">L.G. Horta, J.L. Walsh, G.C. Horner and J.P. Bailey. "[https://ntrs.nasa.gov/search.jsp?R=19870006613 Analysis and simulation of the MAST (COFS-1 flight hardware)]". NASA / DOD Control Structures Interaction Technology: 515--532, 1986.</ref>

</references>

==Contact==

''[[User:Himpe|Christian Himpe]]''

Flexible Aircraft

2025-06-17T05:30:36Z

Schuetz2:

[[Category:benchmark]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:frequency data]]
[[Category:data-driven]]
==Description==

===Motivation===

<figure id="fig:flexibleAC">[[File:Flexible1.png|450px|thumb|right|<caption>The steady pressure coefficient distribution <math>C_p</math> of the aerostructure of the flexible aircraft.</caption>]]</figure>

Flexible aircraft models are very challenging in civil aeronautics due to their light-weight structure.
Models are largely used by engineers to optimize and analyze critical phenomena in the pre-design phase.
Such models can be used to monitor some dimensioning critical stress of the aircraft in response to discrete and continuous gust situations.

Computing responses to discrete gusts are expensive steps when designing and optimizing a new aircraft structure and geometry.
In fact, this is part of the imposed clearance certifications requested by the flight authorities.
During the aircraft preliminary design phase, this clearance is done by intensive simulations.
However, due to the involved model's complexity, these latter are time-consuming and imply an important computational burden.
Moreover, these simulations are involved at different steps of the aircraft optimization process, by aeroelastic, flight and control engineers.

In <ref name=PousstVassal2018>C. Poussot-Vassal, D. Quero, and P. Vuillemin, "[https://doi.org/10.1016/j.ifacol.2018.03.094 Data-driven approximation of a high fidelity gust-oriented flexible aircraft dynamical model]", in Proceedings of the 9th Vienna International Conference on Mathematical Modelling (MATHMOD), Vienna, Austria, 2018, pp 559--564.</ref>, a systematic way to fasten the gust simulation step and simplify the analysis by mean of data-driven model approximation in the Loewner framework is proposed, as well as a description of this model.

===Considered data===

This benchmark contains a set of frequency-domain input-output data computed from a high fidelity simulator, and given as the couple
:<math>
\{\omega_i,H_i^{n_y\times n_u}\}_{i=1}^N,
</math>
where <math>H_i\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (gust disturbance) to <math>n_y=92</math> measurement outputs (accelerations and moments at different coordinates of a flexible aircraft wings and tail), evaluated at varying frequencies <math>\omega_i\in\mathbb R</math> [rad/s], for <math>i=1,\dots,N=421</math>.
Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] and [https://www.onera.fr DLR]. The data from the high fidelity simulator have been generated by D. Quero (DLR), and the treatment performed jointly with P. Vuillemin, D. Quero and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FlexibleAircraft.zip|FlexibleAircraft.zip]] (1.4MB) repository contains three files:

* The <tt>dataONERA_FlexibleAircraft.mat</tt> data file, with
** W : the frequency values in rad/s (real <math>1 \times 421</math> vector).
** H : transfer function matrix evaluation at different output measurements points of the aircraft (complex <math>92 \times 1 \times 421</math> matrix).

* The <tt>dataONERA_FlexibleAircraft_withMOR.mat</tt> data file, with 3 ROMs obtained with the [https://modelreduction.org/morwiki/MOR_Toolbox '''MOR toolbox'''] using the Loewner method
** Hr1 : linear rational ROMs with varying dimensions (state-space models in Matlab form).
** Hr2 : linear rational ROMs with varying dimensions, with stability post enforcement and input/output normalisation to catch all transfer whatever the amplitude (state-space models in Matlab form).

* The <tt>startONERA_FlexibleAircraft.m</tt> script file, used to loads and plots the data for illustration.

The transfer function matrix represents the transfer from the gust input to the 92 measurements gathering from
* 1--44: the local aerodynamic lift on the aerodynamic strips.
* 45--88: the local aerodynamic pitch moment on the aerodynamic strips.
* 89--92: the four generalized coordinates derivative (heave and pitch derivatives) and the first two flexible modes.

===Objective===

Find a (linear) stable reduced order model that well approximates the data.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Flexible Aircraft'''. MORwiki - Model Order Reduction Wiki, 2018. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Flexible_Aircraft

* For the background on the benchmark:

@inproceedings{PoussotMATHMOD:2018,
author = {C. Poussot-Vassal and D. Quero and P. Vuillemin},
title = {Data-driven approximation of a high fidelity gust-oriented flexible aircraft dynamical model},
booktitle = {IFAC PaperOnLine (9th Vienna International Conference on Mathematical Modelling)},
volume = {51},
year = {2018},
pages = {559--564},
doi = {10.1016/j.ifacol.2018.03.094}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Steel Profile

2025-06-17T05:29:18Z

Schuetz2:

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

{{Infobox
|Title = Steel Profile
|Benchmark ID =
* steelProfile_n1357m7q6
* steelProfile_n5177m7q6
* steelProfile_n20209m7q6
* steelProfile_n79841m7q6
|Category = oberwolfach
|System-Class = LTI-FOS
|nstates =
* 1357
* 5177
* 20209
* 79841
|ninputs = 7
|noutputs = 6
|nparameters = 0
|components = A, B, C, E
|License =
|Creator =
* [[User:Saak]]
* Peter Benner
|Editor = [[User:Saak]]
|Zenodo-link = NA
}}

==Description: A Semi-discretized Heat Transfer Problem for Optimal Cooling of Steel Profiles==
<figure id="fig1">[[File:Steelprofile1.jpg|490px|thumb|right|<caption>Initial mesh and partition of the boundary.</caption>]]</figure>
<figure id="fig2">[[File:Steelprofile2.jpg|490px|thumb|right|<caption>Cooling plant.</caption>]]</figure>

A Semi-discretized heat transfer problem for optimal cooling of steel profiles.
Several generalized state-space models arising from a semi-discretization of a controlled heat transfer process for optimal cooling of steel profiles are presented. The models order differs due to different refinements applied to the computational mesh.

===Model Equations===

We consider the problem of optimal cooling of steel profiles.
This problem arises in a [[wikipedia:Rolling_(metalworking)#Mills|rolling mill]] when different steps in the production process require different temperatures of the raw material.
To achieve a high production rate, economical interests suggest to reduce the temperature as fast as possible to the required level before entering the next production phase.
At the same time, the cooling process, which is realized by spraying cooling fluids onto the surface, has to be controlled so that material properties, such as durability or porosity, achieve given quality standards.
Large gradients in the temperature distributions of the steel profile may lead to unwanted deformations, brittleness, loss of rigidity, and other undesirable material properties.
It is therefore the engineer's goal to have a preferably even temperature distribution.

The scientific challenge here is to give the engineers a tool to precalculate different control laws yielding different temperature distributions in order to decide which cooling strategy to choose.

We can only briefly introduce the model here for details we refer to <ref name="Saa03"/>, <ref name="BenS05b"/>, or <ref name="bs04"/>.
We assume an infinitely long steel profile so that we may restrict ourselves to a 2D model.
Exploiting the symmetry of the workpiece, the computational domain <math>\Omega \subset \mathbb{R}^2</math> is chosen as half a cross section of the rail profile.
The heat distribution is modeled by the unsteady linear heat equation on <math>\Omega</math>:

:<math id="eq1">
\begin{align}
c \rho \partial_t x(t,\chi) - \lambda \Delta x(t,\chi) &= 0 \in \mathbb{R}_{>0} \times \Omega, \\
x(0,\chi) &= x_0(\chi) \in \Omega, & (1)\\
\lambda \partial_\nu x(t,\chi) &= g_i \in \mathbb{R}_{>0} \times \Gamma_i,~ \partial \Omega = \bigcup_i \Gamma_i,
\end{align}
</math>

where <math>x</math> is the temperature distribution (<math>x \in H^1([0,\infty],X)</math> with <math>X:=H^1(\Omega)</math> the state space), <math>c=7\,620</math> the specific heat capacity, <math>\lambda=26.4</math> the heat conductivity and <math>\rho=654.0</math> the density of the rail profile.
We split the boundary into several parts <math>\Gamma_i</math> on which we have different boundary functions <math>g_i</math>,
allowing us to vary controls on different parts of the surface.
By <math>\nu</math> we denote the outer normal on the boundary.

We want to establish the control by a feedback law, i.e., we define the boundary functions <math>g_i</math> to be functions of the state <math>x</math> and the control <math>u_i</math>, where <math>(u_i)_i =: u = Fx</math> for a linear operator <math>F</math> which is chosen such that the cost functional

:<math id="eq2">
\begin{align}
J(x_0,u) & := \int_0^\infty (Qy,y)_Y + (Ru,u)_U \operatorname{d}t, & (2)
\end{align}
</math>

with <math>y=Cx</math> is minimized.
Here, <math>Q</math> and <math>R</math> are linear self-adjoint operators on the output space <math>Y</math> and the control space <math>U</math> with <math>Q \geq 0,~ R > 0</math> and <math>C \in L(X,Y)</math>.
The variational formulation of [[#eq1|(1)]] with <math>g_i(t,\chi) = q_i(u_i- x(t,\chi))</math> leads to:

:<math>
(\partial_t x,v) = -\int_\Omega \alpha \nabla x \nabla v \operatorname{d}\chi + \sum_k \Big(q_k u_k \int_{\Gamma_k} (c \rho)^{-1} v \operatorname{d}\sigma - \int_{\Gamma_k} q_k(c\rho)^{-1} xv d\sigma\Big)
</math>

for all <math>v \in C_0^\infty(\Omega)</math>. Here the <math>u_k</math> are the exterior (cooling fluid) temperatures used as the controls,
<math>q_k</math> are constant heat transfer coefficients (i.e., parameters for the spraying intensity of the cooling nozzles) and <math>\alpha := \lambda /(c\rho)</math>.
Note that <math>q_0 = 0</math> gives the Neumann isolation boundary condition on the artificial inner boundary on the symmetry axis.
In view of this weak formulation, we can apply a standard Galerkin approach for discretizing the heat transfer model in space, resulting in a first-order ordinary differential equation. This is described in the following section.

===Discretized Model===

For the discretization we use the <tt>ALBERTA-1.2 fem-toolbox</tt> (see <ref name="SS00"/> for details).
We applied linear Lagrange elements and used a projection method for the curved boundaries.
The initial mesh (see Fig. 1) was produced by MATLABs <tt>pdetool</tt>, which implements a [[wikipedia:Delaunay_triangulation|Delaunay triangulation]] algorithm.
The finer discretizations were produced by global mesh refinement using a bisection refinement method.
The discrete [[wikipedia:Linear–quadratic_regulator|LQR]] problem is then: minimize [[#eq2|(2)]] with respect to:

:<math>
\begin{align}
E \frac{\partial}{\partial t} x(t) &= A x(t) + B u(t), \\
y(t) &= C x(t), \\
x(0) &= x_0,
\end{align}
</math>

with <math>t>0</math>.

==Acknowledgements==

This benchmark example serves as a model problem for the project '''A15: Efficient numerical solution of optimal control problems for instationary convection-diffusion-reaction-equations''' of the Sonderforschungsbereich [https://www.tu-chemnitz.de/sfb393/ SFB393 Parallel Numerical Simulation for Physics and Continuum Mechanics], supported by the [http://www.dfg.de/en/index.jsp Deutsche Forschungsgemeinschaft].
It was motivated by the model described in <ref name="TU01"/>. A very similar problem is used as model problem in the [https://www.tu-chemnitz.de/sfb393/lyapack/ LYAPACK] software package <ref name="Pen00"/>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38881, see <ref name="benner2005"/>.

==Data==

This benchmark includes four different mesh resolutions.
The best FEM-approximation error that one can expect (under suitable smoothness assumptions on the solution) is of order <math>O(h^2)</math> where <math>h</math> is the maximum edge size in the corresponding mesh.
This order should be matched in a model reduction approach.
The following table lists some relevant quantities for the provided models:

{| class="wikitable" style="margin: auto; text-align:right;"
|+ style="caption-side:bottom;"|
|-
|
|# nonzeros in A
|# nonzeros in E
|max. mesh width
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e3-rail_1357.zip SteelProfile-dim1e3-rail_1357.zip] (95kB)
|<math>8\,985</math>
|<math>8\,997</math>
|<math>5.5280 \cdot 10^{-2}</math>
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e4-rail_5177.zip SteelProfile-dim1e4-rail_5177.zip] (299kB)
|<math>35\,185</math>
|<math>35\,241</math>
|<math>2.7640 \cdot 10^{-2}</math>
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e4-rail_20209.zip SteelProfile-dim1e4-rail_20209.zip] (1011kB)
|<math>139\,233</math>
|<math>139\,473</math>
|<math>1.3820 \cdot 10^{-2}</math>
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e5-rail_79841.zip SteelProfile-dim1e5-rail_79841.zip] (3.7MB)
|<math>553\,921</math>
|<math>554\,913</math>
|<math>6.9100 \cdot 10^{-3}</math>
|}

Note that <math>A</math> is negative definite while <math>E</math> is positive definite, so that the resulting linear time-invariant system is stable.

The data sets are named <tt>rail_(problem dimension)_c60.(matrix name)</tt>.
Here, <tt>c60</tt> refers to a specific output matrix <math>C</math>,
which is defined to minimize the temperature in the node numbered 60 (refer to the numbers given in Fig. 1) and keep temperature gradients small.
The latter task is taken into account by the inclusion of temperature differences between specific points in the interior and reference points on the boundary, e.g., temperature differences between nodes 83 and 34.
Again refer to Fig. 1 for the nodes used.
The definitions of other output matrices that we tested can be found in <ref name="Saa03"/>.
The problem resides at temperatures of approximately <math>1\,000</math> degrees centigrade down to about <math>500-700</math> degrees depending on calculation time.
The state values are scaled to <math>1\,000</math> degrees centigrade being equivalent to <math>1.000</math>.
This, together with the scaling of the domain, results in a scaling of the time line with factor <math>100</math>, meaning that calculated times have to be divided by <math>100</math> to get the real time in seconds.

==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 7}</math>,
<math>C \in \mathbb{R}^{6 \times N}</math>

System variants:

<tt>rail1357</tt>: <math>N = 1\,357</math>,
<tt>rail5177</tt>: <math>N = 5\,177</math>,
<tt>rail20209</tt>: <math>N = 20\,209</math>,
<tt>rail79841</tt>: <math>N = 79\,841</math>

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

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

@MISC{morwiki_steel,
author = <nowiki>{{Oberwolfach Benchmark Collection}}</nowiki>,
title = {Steel Profile},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/Steel_Profile}</nowiki>,
year = 2005
}

* For the background on the benchmark:

@TechReport{BenS05b,
title = {Linear-Quadratic Regulator Design for Optimal Cooling of Steel Profiles},
author = {P. Benner and J. Saak},
institution = {Sonderforschungsbereich 393 {\itshape Parallele Numerische Simulation f\"ur Physik und Kontinuumsmechanik}, TU Chem\-nitz},
address = {D-09107 Chem\-nitz (Germany)},
number = {SFB393/05-05},
year = {2005},
url = {<nowiki>http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601597</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], In: Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 311--315, 2005.</ref>

<ref name="benner2005">P. Benner, J. Saak, [https://doi.org/10.1007/3-540-27909-1_19 A Semi-Discretized Heat Transfer Model for Optimal Cooling of Steel Profiles], In: Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol 45: 353--356, 2005.</ref>

<ref name="bs04"> P. Benner, J. Saak, [https://doi.org/10.1002/pamm.200410305 Efficient Numerical Solution of the LQR-problem for the Heat Equation], Proceedings in Applied Mathematics and Mechanics, 4(1): 648--649, 2004.</ref>

<ref name="BenS05b"> P. Benner, J. Saak, [http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601597 Linear-Quadratic Regulator Design for Optimal Cooling of Steel Profiles], Sonderforschungsbereich 393: Parallele Numerische Simulation für Physik und Kontinuumsmechanik, Technical Report SFB393/05-05, TU Chemnitz, 2005.</ref>

<ref name="Pen00"> T. Penzl, [https://www.tu-chemnitz.de/sfb393/Files/PDF/sfb00-33.pdf LYAPACK Users Guide], Sonderforschungsbereich 393: Numerische Simulation auf massiv parallelen Rechnern, Technical Report SFB393/00-33, TU Chemnitz, 2000.</ref>

<ref name="Saa03"> J. Saak, [https://doi.org/10.5281/zenodo.1187041 Effiziente numerische Lösung eines Optimalsteuerungsproblems für die Abkühlung von Stahlprofilen], Diplomarbeit, Fachbereich 3/Mathematik und Informatik, Universität Bremen, 2003.</ref>

<ref name="SS00"> A. Schmidt, K. Siebert, [https://doi.org/10.1007/b138692 Design of Adaptive Finite Element Software - The Finite Element Toolbox ALBERTA], Lecture Notes in Computational Science and Engineering, vol 42, 2005. (See also: [http://www.alberta-fem.de ALBERTA])</ref>

<ref name="TU01"> F. Tröltzsch, A. Unger, [https://doi.org/b2h3hr Fast Solution of Optimal Control Problems in the Selective Cooling of Steel], ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik, 81(7): 447--456, 2001.</ref>

</references>

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

FitzHugh-Nagumo System

2025-06-17T05:28:45Z

Schuetz2:

[[Category:benchmark]]
[[Category:nonlinear]]
[[Category:time invariant]]
[[Category:first differential order]]

==Description==

The [[:Wikipedia:FitzHugh–Nagumo_model|FitzHugh-Nagumo]] system describes a prototype of an excitable system, e.g., a neuron.
If the external stimulus exceeds a certain threshold value, then the system will exhibit a characteristic excursion in phase space, representing activation and deactivation of the neuron. This behaviour is typical for spike generations (=short elevation of membrane voltage) in a neuron after stimulation by an external input current.

<figure id="fig:fhn">
[[File:FHN.png|frame|<caption>FitzHugh-Nagumo System</caption>]]
</figure>

===Model Equations===
Here, we present the setting from <ref name="chat10"/>, where the dynamics of the system is governed by the following coupled nonlinear PDEs:

:<math>
\epsilon v_t(x,t)=\epsilon^2v_{xx}(x,t)+f(v(x,t))-w(x,t)+g,
</math>

:<math>
w_t(x,t)=hv(x,t)-\gamma w(x,t)+g,
</math>

with <math>f(v)=v(v-0.1)(1-v)</math> and initial and boundary conditions

:<math>
v(x,0)=0,\quad w(x,0)=0, \quad x\in [0,1],
</math>

:<math>
v_x(0,t)=-i_0(t), \quad v_x(1,t)\quad =0, \quad t \geq 0,
</math>

where <math>\epsilon=0.015,\;h=0.5,\;\gamma=2,\;g=0.05;\;i_0(t)=5\cdot
10^4t^3 \exp(-15t)</math> is the external stimulus, and the variables <math>v</math> and <math>w</math> are the voltage and the recovery of the voltage, respectively. Fig. 1 shows the typical limit cycle behaviour described above.

==Reformulation as a quadratic-bilinear system==

Instead of the cubic system of ODEs, one can alternatively study a so-called quadratic-bilinear control system of the form

:<math>
E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + \sum_{j=1}^2 N_j x(t) u_j(t) + B u(t),
</math>

with <math>E,A,N_j \in \mathbb R^{n\times n}, j \in\{1,2\}, \ H \in \mathbb R^{n\times n^2} </math> and <math> B \in \mathbb R^{n\times 2}</math>, and the input function is <math>u(t)=[i_0(t),1]</math>. The idea relies on artificially introducing a new state variable defined as <math>z(t)=v(t)^2</math> and subsequently computing the dynamics of the new variable, i.e., specifying <math>\dot{z}(t).</math> The technique goes back to <ref name="gu11"/>, where it is successfully applied to several smooth nonlinear control-affine systems. As discussed in <ref name="benner12"/>, introducing <math>z</math> as an addictional variable yields a quadratic-bilinear control of dimension <math> n = 3\cdot k</math>, where <math>k</math> denotes the number of discretization nodes for each PDE, with state vector <math>x = [v,w,z]^T.</math> The increase of the state dimension has the advantage of reducing the nonlinearity from cubic to quadratic. This, however, opens up the possibility to reduce the system by model reduction schemes developed for quadratic-bilinear systems such as balanced truncation <ref name="morBenG17"/>, or interpolation-based approaches, e.g., given in <ref name="benner12"/> and <ref name="gu11"/>.

==Data==

In <ref name="chat10"/>, the previous system of coupled nonlinear PDEs is spatially discretized by means of a finite difference scheme with <math>k=512 </math> nodes for each PDE. Hence, one obtains a nonlinear (cubic) system of ODEs with state dimension <math>2\cdot k=1024</math>. As discussed above, the nonlinear system can be rewritten as a quadratic-bilinear system of dimension <math>3\cdot k=1536</math>, and all matrices of the quadratic-bilinear formulation discretized are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format. The matrix name is used as an extension of the matrix file and can be found at:

[[Media:FitzHughNagumo.tar.gz|FitzNagumo.tar.gz]].

The archive contains the matrices <math>E</math>, <math>A</math>, <math>B</math>, <math>H</math> and <math>N_1</math>; The matrix <math>N_2</math> is a zero matrix of appropiate size.

For more information on the discretization details, see <ref name="chat10"/>.

In addition, one can have an output
:<math>
y(t) = Cx(t),
</math>
where the matrix <math>C</math> is a <math>2\times N</math> matrix such that <math>C(1,1) = 1,
C(2,1+k) = 1 </math> and the other elements are zero. This output <math>y(t)</math> corresponds to the left boundary of the limit cycles. For more information on the output, see <ref name="benner12"/>.

==Dimensions==

System structure:
:<math>
\begin{align}
E\dot{x}(t) &= A x(t) + H x(t) \otimes x(t) + \sum_{i=1}^2N_i x(t) u_i(t) + B u(t) \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>E \in \mathbb{R}^{1\,536 \times 1\,536}</math>,
<math>A \in \mathbb{R}^{1\,536 \times 1\,536}</math>,
<math>H \in \mathbb{R}^{1\,536 \times 2\,359\,296}</math>,
<math>N_1 \in \mathbb{R}^{1\,536 \times 1\,536}</math>,
<math>N_2 \in \mathbb{R}^{1\,536 \times 1\,536}</math>,
<math>B \in \mathbb{R}^{1\,536 \times 2}</math>,
<math>C \in \mathbb{R}^{2 \times 1\,536}</math>.

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

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

@MISC{morwiki_modFHN,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {FitzHugh-Nagumo System},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/FitzHugh-Nagumo_System}</nowiki>,
year = 2018
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morChaS10 morChaS10] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#ChaS10 BibTeX])

==References==

<references>

<ref name="chat10">S. Chaturantabut and D.C. Sorensen, "[https://doi.org/10.1137/090766498 Nonlinear Model Reduction via Discrete Empirical Interpolation]", SIAM J. Sci. Comput., 32: 2737--2764, 2010.</ref>

<ref name="gu11">C. Gu, "[https://doi.org/10.1109/TCAD.2011.2142184 QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems]", IEEE T. Comput. Aid. D., 30: 1307--1320, 2011.</ref>

<ref name="benner12">P. Benner and T. Breiten, "[https://doi.org/10.1137/14097255X Two-Sided Moment Matching Methods for Nonlinear Model Reduction]", SIAM J. Sci. Comput., 37(2): B239--B260, 2015.</ref>

<ref name="morBenG17">P. Benner and P. Goyal, "[https://arxiv.org/abs/1705.00160 Balanced Truncation Model Order Reduction for Quadratic-Bilinear Systems]", arXiv e-prints, 2017</ref>

</references>

==Contact==

'' [[User:Breiten]]''

Electrostatic Beam

2025-06-12T05:28:27Z

Schuetz2:

[[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>{https://modelreduction.org/morwiki/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>

Coplanar Waveguide

2025-06-12T05:27:25Z

Schuetz2:

[[Category:benchmark]]
[[Category:PDE]]
[[Category:Parametric]]
[[Category:linear]]
[[Category:affine parameter representation]]
[[Category:Stationary]]

==Description==

A '''coplanar waveguide''' (see Fig. 1) is a microwave semiconductor device, which is governed by [[wikipedia:Maxwell's_equations|Maxwell's equations]].
The [[wikipedia:Coplanar_waveguide|coplanar waveguide]] considered with dielectric overlay, i.e. a transmission line shielded within two layers of multilayer board with <math>0.5 \, \text{mm}</math> thickness are buried in a substrate with <math>10 \, \text{mm}</math> thickness and relative permittivity
<math>\epsilon_r = 4.4</math> and relative permeability <math>\mu_r = 1</math>, and low conductivity <math>\sigma = 0.02 \, \text{S/m}</math>.
The low-loss upper layer has low permittivity <math>\epsilon_r = 1.07</math> and <math>\sigma = 0.01 \, \text{S/m}</math>.
The whole structure is enclosed in a metallic box of dimension <math>140 \, \text{mm}</math> by <math>100 \, \text{mm}</math> by <math>50 \, \text{mm}</math>.
The discrete port with <math>50 \, \Omega</math> lumped load imposes <math>1 \, \text{A}</math> current as the input to the one side of the strip.
The voltage along the discrete port 2 at the end of the other side of coupled lines is integrated as the output.

<figure id="fig:coplanar">
[[File:CoplanarWaveguideScaled.jpg|frame|<caption>Coplanar Waveguide Model<ref name="hess2003"/></caption>]]
</figure>

==Data==

Considered parameters are the frequency <math>\omega</math> and the width <math>\nu</math> of the middle stripline.

The affine form <math>a(u, v; \omega, \nu) = \sum_{q=1}^Q \Theta^q(\omega, \nu) a^q(u, v)</math> can be established using <math>Q = 15</math> affine terms.

The discretized bilinear form is <math>a(u, v; \omega, \nu) = \sum_{q=1}^Q \Theta^q(\omega, \nu) A^q</math>, with matrices <math>A^q</math>.

The matrices corresponding to the bilinear forms <math>a^q(\cdot, \cdot)</math> as well as the input and output forms and H(curl) inner product matrix have been assembled
using the Finite Element Method, resulting in 7754 degrees of freedom, after removal of boundary conditions. The files are numbered according to their
appearance in the summation.

The coefficient functions are given by:

:<math>
\begin{align}
\Theta^1(\omega, \nu) &= 1, \\
\Theta^2(\omega, \nu) &= \omega, \\
\Theta^3(\omega, \nu) &= -\omega^2, \\
\Theta^4(\omega, \nu) &= \frac{\nu}{6}, \\
\Theta^5(\omega, \nu) &= \frac{6}{\nu}, \\
\Theta^6(\omega, \nu) &= \frac{6 \omega}{\nu}, \\
\Theta^7(\omega, \nu) &= -\frac{6 \omega^2}{\nu}, \\
\Theta^8(\omega, \nu) &= \frac{\nu \omega}{6}, \\
\Theta^9(\omega, \nu) &= -\frac{\nu \omega^2}{6}, \\
\Theta^{10}(\omega, \nu) &= \frac{16 - \nu}{10}, \\
\Theta^{11}(\omega, \nu) &= \frac{10}{16 - \nu}, \\
\Theta^{12}(\omega, \nu) &= \frac{10 \omega}{16 - \nu}, \\
\Theta^{13}(\omega, \nu) &= -\frac{10 \omega^2}{16 - \nu}, \\
\Theta^{14}(\omega, \nu) &= \frac{16 - \nu}{10} \omega, \\
\Theta^{15}(\omega, \nu) &= -\frac{16 - \nu}{10} \omega^2.
\end{align}
</math>

The parameter domain of interest is <math>\omega \in [0.6, 3.0] \cdot 10^9 \, \text{Hz}</math>, where the factor of <math>10^9</math> has already been taken into account
while assembling the matrices, while the geometric variation occurs between <math>\nu \in [2.0, 14.0]</math>.
The input functional also has a factor of <math>\omega</math>.

There are two output functionals, which is due to the fact that the complex system has been rewritten as a real symmetric one.
In particular the computation of the output
<math>s(u) = \vert l^T u \vert</math> with complex vector <math>u</math> turns into <math>s(u) = \sqrt{(l_1^T u)^2 + (l_2^T u)^2}</math> with real vector <math>u</math>.

==Origin==

The models have been developed within the [http://www.moresim4nano.org MoreSim4Nano] project.

==Data==

The files are numbered according to their appearance in the summation and can be found here: [[Media:Matrices_cp.tar.gz|Matrices_cp.tar.gz]].

==Dimensions==

System structure:

:<math>
\begin{align}
\sum_{q = 1}^{15} \Theta^q(\omega, \nu) A^q u(\omega, \nu) &= b \\
s(\omega, \nu) &= \sqrt{(l_1^T u(\omega, \nu))^2 + (l_2^T u(\omega, \nu))^2}
\end{align}
</math>

System dimensions:

<math>A^q \in \mathbb{R}^{15504 \times 15504}</math>,
<math>b, l_1, l_2 \in \mathbb{R}^{15504}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_waveguide,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Coplanar Waveguide},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/Coplanar_Waveguide}</nowiki>,
year = {2018}
}

* For the background on the benchmark:
@ARTICLE{morHesB13,
author = {Hess, M.~W. and Benner, P.},
title = {Fast Evaluation of Time-Harmonic {M}axwell's Equations Using the Reduced Basis Method},
journal = {{IEEE} Trans. Microw. Theory Techn.},
volume = 61,
number = 6,
pages = {2265--2274},
year = 2013,
doi = {10.1109/TMTT.2013.2258167}
}

==References==

<references>

<ref name="hess2003">M.W. Hess, P. Benner, "[https://doi.org/10.1109/TMTT.2013.2258167 Fast Evaluation of Time-Harmonic Maxwell's Equations Using the Reduced Basis Method]", IEEE Transactions on Microwave Theory and Techniques, 61(6): 2265--2274, 2013.</ref>

</references>

==Contact==

'' [[User:hessm|Martin Hess]]''

Convective Thermal Flow

2025-06-12T05:26:55Z

Schuetz2:

[[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>{https://modelreduction.org/morwiki/Convection}</nowiki>,
year = {2018}
}

* 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>

Circular Piston

2025-06-12T05:24:12Z

Schuetz2:

[[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>{https://modelreduction.org/morwiki/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>

Batch Chromatography

2025-06-11T08:59:17Z

Schuetz2:

[[Category:benchmark]]
[[Category:PDE]]
[[Category:nonlinear]]
[[Category:Parametric]]

==Description==

<figure id="fig:bach">[[File:Fig_BatchChrom.jpg|490px|thumb|right|<caption>Sketch of a batch chromatographic process for the separation of A and B.</caption>]]</figure>

Preparative liquid chromatography as a crucial separation and purification tool has been widely employed in the food, fine chemical, and pharmaceutical industries. Chromatographic separation at the industry scale can be operated either discontinuously or in a continuous mode.
The continuous case will be discussed in the benchmark [[SMB]], and here we focus on the discontinuous mode -- [[wikipedia:Affinity_chromatography#Batch_and_column_setups|batch chromatography]].
The principle of the batch chromatographic process for the binary separation is shown in Fig. 1.
During the injection period <math>t_{inj}</math>, a mixture of products A and B is injected at the inlet of the column packed with a suitable stationary phase.
With the help of the mobile phase, the feed mixture flows through the column.
Since the to-be-separated solutes exhibit different adsorption affinities to the stationary phase, they move at different velocities
and thus separate from each other when exiting the column.
At the column outlet, component A is collected between <math>t_1</math> and <math>t_2</math>, and component B is collected between <math>t_3</math> and <math>t_4</math>.
Here the positions of <math>t_1</math> and <math>t_4</math> are determined by a minimum concentration threshold that the detector can resolve, and the positions of <math>t_2</math> and <math>t_3</math> are determined by the purity specifications imposed on the products.
After the cycle period <math>t_{cyc}:=t_4-t_1</math>, the injection is repeated.
The feed flow rate <math>Q</math> and injection period <math>t_{inj}</math> are often considered as the operating variables.
By properly choosing them, the process can achieve the desired performance criterion, such as production rate, while respecting the product specifications (e.g., purity, recovery yield).

The dynamics of the batch chromatographic column can be described precisely by an axially dispersed plug-flow model with a limited mass-transfer rate characterized by a linear driving force (LDF) approximation.
In this model the differential mass balance of component <math>i</math> (<math>i=A,B,</math>) in the liquid phase can be written as:

:<math>
\frac{\partial c_i}{\partial t}
+ \frac{1 - \epsilon}{\epsilon} \frac{\partial q_i}{\partial t}
+ u \frac{\partial c_i}{\partial z}
- D_i \frac{\partial^2 c_i}{\partial z^2}
= 0, \qquad
z \in (0, L), \qquad (1)
</math>

where <math>c_i</math> and <math>q_i</math> are the concentrations of solute <math>i</math> in the liquid and solid phases, respectively, <math>u</math> the interstitial liquid velocity, <math>\epsilon</math> the column porosity, <math>t</math> the time coordinate, <math>z</math> the axial coordinate along the column, <math>L</math> the column length, <math>D_i=\frac{uL}{Pe}</math> the axial dispersion coefficient and <math>Pe</math> the [[wikipedia:Péclet_number|Péclet number]].
The adsorption rate is modeled by the LDF approximation:

:<math>
\frac{\partial q_i}{\partial t}
= \kappa_{i} \left(q^{Eq}_i - q_i\right), \qquad
z \in [0, L],
</math>

where <math>\kappa_{i}</math> is the mass-transfer coefficient of component <math>i</math> and <math>q^{Eq}_i</math> is the adsorption equilibrium concentration calculated by the isotherm equation for component <math>i</math>. Here the bi-[[wikipedia:Langmuir_adsorption_model|Langmuir]] isotherm model is used to describe the adsorption equilibrium:

:<math>
q^{Eq}_i
= \frac{H_{i,1}\,c_i}{1 + \sum_{j = A, B}K_{j,1}\,c_j}
+ \frac{H_{i,2}\,c_i}{1 + \sum_{j = A, B}K_{j,2}\,c_j},\;
i = A, B,
</math>

where <math>H_{i,1}</math> and <math>H_{i,2}</math> are the Henry constants, and <math>K_{j,1}</math> and <math>K_{j,2}</math> the thermodynamic coefficients.

The boundary conditions for (1) are specified by the Danckwerts relations:

:<math>
D_i \left.\frac{\partial c_i}{\partial z}\right|_{z = 0}
= u \left(\left.c_i\right|_{z=0} - c^{in}_i\right), \qquad
\left.\frac{\partial c_i}{\partial z}\right|_{z = L}
= 0,
</math>

where <math>c^{in}_i</math> is the concentration of component <math>i</math> at the inlet of the column.
A rectangular injection is assumed for the system and thus

:<math>
c^{in}_i
=
\begin{cases}
c^F_i, & \text{if } t \le t_{inj}, \\
0, & \text{if } t > t_{inj}.
\end{cases}
</math>

Here <math>c^F_i</math> is the feed concentration for component <math>i</math> and <math>t_{inj}</math> is the injection period.
In addition, the column is assumed unloaded initially:

:<math>
c_i(t = 0, z)
= q_i(t = 0,z)
= 0, \quad
z \in [0, L], \;
i = A, B.
</math>

More details about the mathematical modeling for batch chromatography can be found in the literature <ref name="guiochon06"/>.

==Discretization==
In this model, the feed volumetric flow-rate <math>Q</math> and the injection period <math>t_{inj}</math> are considered as the operating parameters, and denoted as the parameter <math>\mu=(Q,\,t_{inj})</math>. Using the finite volume discretization, we get the full order model (FOM) as follows,

:<math>
\left\{
\begin{aligned}
\mathbf{A} \mathbf{c}_i^{k+1}
&= \mathbf{B} \mathbf{c}_i^{k}
+ d_i^k
- \frac{1 - \epsilon}{\epsilon} \Delta t \mathbf{h}_i^k, \\
\mathbf{q}_i^{k + 1}
&= \mathbf{q}_i^{k}
+ \Delta t \mathbf{h}_i^k,
\end{aligned}
\right.
</math>

where <math>\mathbf{c}_i^k, \mathbf{q}_i^k \in \mathbb{R}^{\mathcal N}, i = A, B</math> are the solution vectors of <math>c_i</math> and <math>q_i</math> at the time instance <math>t = t^k, k = 0, 1, \ldots, K</math>, respectively.
The time step <math>\Delta t </math> is determined by the stability condition.
The equation <math> \mathbf h_i^k = \kappa_{i} (\mathbf q^{Eq}_i - \mathbf q_i^k) </math>, is time- and parameter-dependent, the boldface <math>\mathbf{A,B}</math> are constant matrices. As a result, it is a nonlinear parametric system.

==Generation of ROM==

The reduced order model (ROM) can be obtained by the reduced basis method <ref name="zhang15"/>, which is applicable for nonlinear parametric systems, see
[[Reduced_Basis_PMOR_method|Reduced Basis PMOR method]].
For parametrized time-dependent problems, the reduced basis is often generated by using the POD-Greedy algorithm <ref name="haasdonk08"/>.
Notice that the nonlinear functions <math>\mathbf h_i, i=A, B</math> can be approximated by the empirical interpolation method <ref name="barrault04"/>,
such that the ROM can be obtained efficiently by the strategy of offline-online decomposition.

Assume <math>W_z</math> is the collateral reduced basis (CRB) for the nonlinear operator <math>h_z</math>, and <math>V_{c_z},V_{q_z}</math> are the reduced bases for the field variables <math>c_z</math> and <math>q_z</math>, respectively.
Applying Galerkin projection and empirical operator interpolation, the ROM can be formulated as:

:<math>
\left\{
\begin{aligned}
\hat{A}_{c_z} {a}_{c_z}^{n+1}
&= \hat{B}_{c_z} {a}_{c_z}^{n}
+ d_0^n \hat{d}_{c_z}
- \frac{1 - \epsilon}{\epsilon} \Delta t \hat{H}_{c_z} \beta_z^n, \\
{a}_{q_z}^{n + 1}
&= {a}_{q_z}^{n}
+ \Delta t \hat{H}_{q_z} \beta_z^n,
\end{aligned}
\right.
</math>

where <math> {a}_{c_z}^n, {a}_{q_z}^n \in \mathbb R^N</math> are the solution of the ROM. <math>\hat{A}_{c_z}=V_{c_z}^T A V_{c_z},
\hat{ B}_{c_z}=V_{c_z}^T B V_{c_z},
\hat{d}_{c_z}^{n}=V_{c_z}^T e_1</math>,
<math>\hat{H}_{c_z}:= V_{c_z}^TW_z ,</math>
<math>\hat{H}_{q_z}:= V_{q_z}^TW_z </math> are the reduced matrices, <math>e_1:=(1,0,\cdots,0)</math>.
<math>\beta_z^n \in \mathbb R^M</math> is the coefficients of the CRB <math>W_z</math> for the empirical interpolation.

==Data==
<figure id="fig:cabfr">[[File:cabfrA.png|480px|thumb|right|<caption>Concentrations at the outlet of the column.</caption>]]</figure>

Fig. 2 shows the concentrations at the outlet of the column at a given parameter <math>\mu= (0.1018, 1.3487)</math>, which show that the ROM (<math>N=46, M=151</math>) reproduces the dynamics of the full order model (<math>\mathcal N=1000</math>).

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_bone,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Batch Chromatography},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/index.php/Batch_Chromatography}</nowiki>,
year = {2018}
}

* For the background on the benchmark:

@ARTICLE{RieWHetal95,
author = <nowiki>{Y. Zhang and L. Feng and S. Li and P. Benner}</nowiki>,
title = {Accelerating {PDE} constrained optimization by the reduced basis method: application to batch chromatography},
journal = {International Journal for Numerical Methods in Engineering},
volume = {104},
number = {11},
pages = {983--1007},
year = {2015},
doi = {10.1002/nme.4950}
}

==References==

<references>

<ref name="guiochon06"> G. Guiochon, A. Felinger, D. G. Shirazi, A. M. Katti, [http://books.google.com/books/about/Fundamentals_of_preparative_and_nonlinea.html?id=UjZRAAAAMAAJ Fundamentals of Preparative and Nonlinear Chromatography], 2nd Edition, Academic Press, 2006.</ref>

<ref name="barrault04"> M. Barrault, Y. Maday, N.C. Nguyen, and A.T. Patera, "[https://doi.org/10.1016/j.crma.2004.08.006 An 'empirical interpolation' method: application to efficient reduced-basis discretization of partial differential equations]", C. R. Acad. Sci. Paris Series I, 339 (2004), 667-672.</ref>

<ref name="haasdonk08"> B. Haasdonk and M. Ohlberger, "[https://doi.org/10.1051/m2an:2008001 Reduced basis method for finite volume approximations of parameterized linear evolution equations]", Mathematical Modeling and Numerical Analysis, 42 (2008), 277-302.</ref>

<ref name="zhang15"> Y. Zhang, L. Feng, S. Li and P. Benner, "[https://doi.org/10.1002/nme.4950 Accelerating PDE constrained optimization by the reduced basis method: application to batch chromatography]", International Journal for Numerical Methods in Engineering, 104(11): 983--1007, 2015.</ref>

</references>

==Contact==

'' [[User:Zhangy|Yongjin Zhang]]''

'' [[User:Suzhou|Suzhou Li]]''

Anemometer

2025-06-11T08:42:50Z

Schuetz2:

[[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>{https://modelreduction.org/morwiki/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]]

All pass system

2025-06-11T08:42:31Z

Schuetz2:

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

==Description==

This procedural benchmark generates an all-pass SISO system based on <ref name="Obe87"/>.
For an all-pass system, the transfer function has the property <math>g(s)g(-s) = \sigma^2</math>, <math>\sigma > 0</math>,
or (equivalently) the controllability and observability Gramians are quasi inverse to each other: <math>W_C W_O = \sigma I</math>,
which means this system has a single Hankel singular value of multiplicity of the system's order.
The system matrices are constructed based on the scheme:

:<math>
\begin{align}
A &=
\begin{pmatrix}
a_{1,1} & -\alpha_1 \\
\alpha_1 & 0 & -\alpha_2 \\
& \alpha_2 & 0 & \ddots \\
& & \ddots & \ddots & -\alpha_{N-1} \\
& & & \alpha_{N-1} & 0
\end{pmatrix}, \\
B &=
\begin{pmatrix}
b_1 \\
0 \\
\vdots \\
0
\end{pmatrix}, \\
C &=
\begin{pmatrix}
s_1 b_1 & 0 & \cdots & 0
\end{pmatrix}, \\
D &= -s_1 \sigma.
\end{align}
</math>

We choose <math>s_1 \in \{-1,1\}</math> to be <math>s_1 \equiv -1</math>, as this makes the system state-space-anti-symmetric.
Furthermore, <math>b_1 = 1</math> and <math>\sigma = 1</math>, which makes <math>a_{1,1} = -\frac{b_1^2}{2 \sigma} = -\frac{1}{2}</math>.

==Data==

This benchmark is procedural and the state dimensions can be chosen.
Use the following [http://matlab.com MATLAB] code to generate a random system as described above:


:<syntaxhighlight lang="matlab">
function [A,B,C,D] = allpass(n)
% allpass (all-pass system)
% by Christian Himpe, 2020
% released under BSD 2-Clause License
%*

A = gallery('tridiag',n,-1,0,1);
A(1,1) = -0.5;
B = sparse(1,1,1,n,1);
C = -B';
D = 1;
end
</syntaxhighlight>

The function call requires one argument; the number of states <math>n</math>.
The return value consists of four matrices; the system matrix <math>A</math>, the input matrix <math>B</math>, the output matrix <math>C</math>, and the feed-through matrix <math>D</math>.

:<syntaxhighlight lang="matlab">
[A,B,C,D] = allpass(n);
</syntaxhighlight>

An equivalent [https://www.python.org/ Python] code is

:<syntaxhighlight lang="python">
from scipy.sparse import diags, lil_matrix

def allpass(n):
A = diags([-1, 0, 1], offsets=[-1, 0, 1], shape=(n, n), format='lil')
A[0, 0] = -0.5
A = A.tocsc()
B = lil_matrix((n, 1))
B[0, 0] = 1
B = B.tocsc()
C = -B.T
D = 1
return A, B, C, D
</syntaxhighlight>

==Dimensions==

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

System dimensions:

<math>A \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>,
<math>D \in \mathbb{R}</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''All-Pass System'''. MORwiki - Model Order Reduction Wiki, 2020. http://modelreduction.org/index.php/All_pass_system

@MISC{morwiki_allpass,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {All-Pass System},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{https://modelreduction.org/morwiki/index.php/All_pass_system}</nowiki>,
year = {2020}
}

==References==

<references>

<ref name="Obe87">R.J. Ober. "[https://doi.org/10.1016/S1474-6670(17)55030-2 Asymptotically Stable All-Pass Transfer Functions: Canonical Form, Parametrization and Realization]", IFAC Proceedings Volumes, 20(5): 181--185, 1987.</ref>

</references>

Butterfly Gyroscope

2025-06-11T08:41:54Z

Schuetz2:

[[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>{https://modelreduction.org/morwiki/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>