MAX stand - Revision history

Schuetz2 at 07:39, 3 March 2026

2026-03-03T07:39:23Z

← Older revision		Revision as of 07:39, 3 March 2026
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	}}		}}

	~~__NUMBEREDHEADINGS__~~

	==Motivation==		==Motivation==

Vettermann: add citation

2023-10-26T12:37:00Z

add citation

← Older revision		Revision as of 12:37, 26 October 2023
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	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.		The '''MAX stand''' is used as a demonstrator in the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Category:CRC-TR-96 CRC/TR 96] Project-ID 174223256 financed by the German Research Foundation DFG.

			==Citation==

			To cite this benchmark, use the following references:

			* For the benchmark itself and its data:

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

	==References==		==References==

Vettermann: Add Infobox and Zenodo link

2023-10-25T14:08:20Z

Add Infobox and Zenodo link

@@ Line 9: / Line 9: @@
 [[Category:MIMO]]
 [[Category:CRC-TR-96]]
 __NUMBEREDHEADINGS__
@@ Line 75: / Line 114: @@
 :<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\\
+  E_k\dot{T}_k&=(A_k-\sum_{i=1}^{l_k}\kappa_{ext_i}(t)A^i_k)T_k+B_k u_k(t),\quad k=1,\dots,4\\
   y_k(t)&=C_k T_k,\\
 \end{align}
@@ Line 85: / Line 124: @@
 | <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> A_k \in \mathbb{R}^{n\times n} </math> || - basic stiffness matrix (discrete Laplacian)
 |-
 | <math> l_k \in \mathbb{R} </math> || - number of boundary conditions with the ambience (Robin boundary conditions (RBC))
 |-
-| <math> E^{\Gamma}_k \in \mathbb{R}^{n\times n} </math> || - boundary mass matrix (part of stiffness matrix arising from RBC)
+| <math> A^i_k \in \mathbb{R}^{n\times n} </math> || - boundary mass matrix (part of stiffness matrix arising from RBC)
 |-
 | <math> T_k \in \mathbb{R}^n </math> || - state vector (discrete temperature)
@@ Line 109: / Line 148: @@
 ==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
+The system matrices (heat transfer coefficients with the ambience may be varied) and files for a simulation of the model in time domain can be downloaded from [https://zenodo.org/records/10039696 Zenodo]. There is a set of matrices for each subassembly. The numbering of the subassemblies is shown in Fig. 1. Further the archive contains a MATLAB file with the description and computation of the contacts between the subassemblies and a file for the simulation of a work process. The overall system dimension is <math> n=83\,763 </math> with
 {| class="wikitable" style="margin: auto;"

Vettermann: add link to data

2023-06-09T14:29:47Z

add link to data

← Older revision		Revision as of 14:29, 9 June 2023
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	==Data==		==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		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;"		{\| class="wikitable" style="margin: auto;"

Vettermann: add a remark on the shift in the simulation

2023-06-09T05:48:11Z

add a remark on the shift in the simulation

← Older revision		Revision as of 05:48, 9 June 2023
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	==Data==		==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		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;"		{\| class="wikitable" style="margin: auto;"
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	* This is a model with diagonal matrix <math>E</math>.		* This is a model with diagonal matrix <math>E</math>.
	* The model was also used for numerical experiments presented in <ref name="palitta2023"/>.		* 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==		==Origin==

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2023-06-06T14:33:04Z

add comment on missing link to the data

← Older revision		Revision as of 14:33, 6 June 2023
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	==Data==		==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		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;"		{\| class="wikitable" style="margin: auto;"

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

2023-06-06T14:25:17Z

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

New page

{{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]]''