MOR Wiki - User contributions [en]

Battery pack

2024-03-15T14:39:08Z

Vettermann: add DOI

{{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
}}

__NUMBEREDHEADINGS__

== 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>

Battery pack

2024-03-15T14:37:18Z

Vettermann: add zenodo link

{{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
}}

__NUMBEREDHEADINGS__

== 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 = {}
}

== 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>

Battery pack

2024-03-12T08:39:02Z

Vettermann: Created page with "{{preliminary}}  Category:benchmark Category:parametric Category:sparse Category:linear Category:first differential order Category..."

{{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 = N.A.
|Creator = Lucas Kostetzer ([http://www.cadfem.de/ CADFEM])
|Editor = [[User:Vettermann]]
|Zenodo-link = N.A.
}}

__NUMBEREDHEADINGS__

== 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 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},
title = {Battery pack},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2024,
doi = {}
}

== 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>

2023-09-05T06:24:22Z

Vettermann:

{{preliminary}} 

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

__NUMBEREDHEADINGS__

==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 under the following link [https://tuc.cloud/index.php/s/diyz2JNzjkkt9RA MAX.tar.gz]. The model has <math> n=1\,265\,497 </math> degrees of freedom and is available for two diffenrent 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 and thus the subassemly models can be reduced separately for structure preserving MOR. There is a set of matrices for each subassembly with a total number of <math> m=287 </math> inputs and <math> p=224 </math> outputs.
* 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.

==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, J. Saak, P. Benner, M. Beitel-
schmidt and R. Herzog, "[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]]''

Machine tool MAX

2023-09-04T12:17:17Z

Vettermann: Created page with "{{preliminary}}  Category:benchmark Category:Sparse Category:linear Category:first differential order Category:ODE Category:MIMO..."

{{preliminary}} 

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

__NUMBEREDHEADINGS__

==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,4, \\
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, 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> \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 under the following link [https://tuc.cloud/index.php/s/diyz2JNzjkkt9RA MAX.tar.gz]. The model has <math> n=1\,265\,497 </math> degrees of freedom and is available for two diffenrent 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 and thus the subassemly models can be reduced separately for structure preserving MOR. There is a set of matrices for each subassembly with a total number of <math> m=287 </math> inputs and <math> p=224 </math> outputs.
* 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.

==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, J. Saak, P. Benner, M. Beitel-
schmidt and R. Herzog, "[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]]''

File:Coupled systems.pdf

2023-09-04T12:06:10Z

Vettermann:

File:MAX real.jpg

2023-09-04T12:05:36Z

Vettermann:

File:MAX FE.pdf

2023-09-04T12:04:19Z

Vettermann:

Simplified machine tool

2023-09-01T15:07:31Z

Vettermann: Created page with "{{preliminary}}  Category:benchmark Category:Sparse Category:linear Category:first differential order Category:ODE Category:MIMO..."

{{preliminary}} 

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

__NUMBEREDHEADINGS__

==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 under the following link [https://tuc.cloud/index.php/s/apdEG37gj8EaRDY MAX_stand.tar.gz]. 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.

==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]]''

File:DMU Dummy subassemblies.png

2023-09-01T14:55:45Z

Vettermann:

Vertical Stand

2023-06-20T05:45:04Z

Vettermann: corrected heat equation

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

__NUMBEREDHEADINGS__

==Description==

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

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

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

with the boundary conditions

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

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

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

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

'''Geometrical dimensions:'''

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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

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

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

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

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



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

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

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

==References==

<references>

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

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

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

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

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

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

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

</references>

==Contact==

'' [[User:Saak]]''

MAX stand

2023-06-09T14:29:47Z

Vettermann: add link to data

{{preliminary}} 

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

__NUMBEREDHEADINGS__

==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_{0_k}-\sum_{i=1}^{l_k}\kappa_{ext_i}(t)E^\Gamma_{k_i})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_{0_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> E^{\Gamma}_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 as well as scripts to sample the conductivity matrix A (heat transfer coefficients with the ambience may be varied) and for a simulation of the model in time domain can be found in [https://tuc.cloud/index.php/s/nrgabQEj3jotrst MAX_stand.tar.gz]. There is a set of matrices for each subassembly. The numbering of the subassemblies is shown in Fig. 1. Further the archive contains some MATLAB files with the description and computation of the contacts between the subassemblies. 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.

==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]]''

MAX stand

2023-06-09T05:48:11Z

Vettermann: add a remark on the shift in the simulation

{{preliminary}} 

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

__NUMBEREDHEADINGS__

==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_{0_k}-\sum_{i=1}^{l_k}\kappa_{ext_i}(t)E^\Gamma_{k_i})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_{0_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> E^{\Gamma}_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 as well as scripts to sample the conductivity matrix A (heat conductivity coefficients may be varied) and for a simulation of the model in time domain can be found in [ MAX_stand.tar.gz](link will be added on June 9th). There is a set of matrices for each subassembly. The numbering of the subassemblies is shown in Fig. 1. Further the archive contains some MATLAB files with the description and computation of the contacts between the subassemblies. 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.

==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]]''

MAX stand

2023-06-06T14:33:04Z

Vettermann: add comment on missing link to the data

{{preliminary}} 

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

__NUMBEREDHEADINGS__

==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_{0_k}-\sum_{i=1}^{l_k}\kappa_{ext_i}(t)E^\Gamma_{k_i})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_{0_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> E^{\Gamma}_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 as well as scripts to sample the conductivity matrix A (heat conductivity coefficients may be varied) and for a simulation of the model in time domain can be found in [ MAX_stand.tar.gz](link will be added on June 7th). There is a set of matrices for each subassembly. The numbering of the subassemblies is shown in Fig. 1. Further the archive contains some MATLAB files with the description and computation of the contacts between the subassemblies. 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"/>.

==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.

==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]]''

File:MAX stand.png

2023-06-06T14:28:05Z

Vettermann:

MAX stand

2023-06-06T14:25:17Z

Vettermann: Created page with "{{preliminary}}  Category:benchmark Category:Parametric Category:Sparse Category:linear Category:first differential order Category..."

{{preliminary}} 

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

__NUMBEREDHEADINGS__

==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_{0_k}-\sum_{i=1}^{l_k}\kappa_{ext_i}(t)E^\Gamma_{k_i})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_{0_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> E^{\Gamma}_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 as well as scripts to sample the conductivity matrix A (heat conductivity coefficients may be varied) and for a simulation of the model in time domain can be found in [ MAX_stand.tar.gz]. There is a set of matrices for each subassembly. The numbering of the subassemblies is shown in Fig. 1. Further the archive contains some MATLAB files with the description and computation of the contacts between the subassemblies. 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"/>.

==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.

==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]]''

2023-05-15T10:33:20Z

Vettermann: Created page with "{{preliminary}}  Category:benchmark The [https://transregio96.webspace.tu-dresden.de/index.php/thermo-energetic-design-of-machine-tools/ Collaborati..."

{{preliminary}} 
[[Category:benchmark]]

The [https://transregio96.webspace.tu-dresden.de/index.php/thermo-energetic-design-of-machine-tools/ Collaborative Research Centre/ Transregio 96] deals with the topic of thermo-energetic design of machine tools.

The challenge of research in the CRC/TR 96 derives from the attempt to satisfy the conflicting goals of reducing energy consumption and increasing accuracy and productivity in machining. The solution approach pursued is based on measures that make it possible to guarantee process accuracy despite increasing power losses without additional energetic measures under thermal transient environmental conditions and under operating conditions characterized by individual and small series production. The scientists of the CRC/TR 96 are researching and developing effective correction and compensation solutions for the thermo-elastic machine behavior during the course of the project, which will enable precision machining under the future conditions of energy-efficient production.

The CRC/TR 96 benchmark collection contains benchmarks related to thermo-elastic machine tool models which were used for the development and application of model order reduction techniques.

Vertical Stand

2023-05-15T10:26:20Z

Vettermann:

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

__NUMBEREDHEADINGS__

==Description==

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

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

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

with the boundary conditions

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

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

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

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

'''Geometrical dimensions:'''

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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

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

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

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

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



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

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

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

==References==

<references>

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

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

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

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

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

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

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

</references>

==Contact==

'' [[User:Saak]]''