Steel Profile - Revision history

Schuetz2 at 05:29, 17 June 2025

2025-06-17T05:29:18Z

← Older revision		Revision as of 05:29, 17 June 2025
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	title = {Steel Profile},		title = {Steel Profile},
	howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},		howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
	url = <nowiki>{~~http~~://modelreduction.org/~~index.php~~/Steel_Profile}</nowiki>,		url = <nowiki>{https://modelreduction.org/morwiki/Steel_Profile}</nowiki>,
	year = 2005		year = 2005
	}		}

Schulze: Fix file names within data sets

2024-04-11T12:35:51Z

Fix file names within data sets

← Older revision		Revision as of 12:35, 11 April 2024
Line 159:		Line 159:
	Note that <math>A</math> is negative definite while <math>E</math> is positive definite, so that the resulting linear time-invariant system is stable.		Note that <math>A</math> is negative definite while <math>E</math> is positive definite, so that the resulting linear time-invariant system is stable.

	The data sets are named <tt>rail_(problem dimension)~~_C60~~.(matrix name)</tt>.		The data sets are named <tt>rail_(problem dimension)_c60.(matrix name)</tt>.
	Here <tt>~~C60~~</tt> is a specific output matrix which is defined to minimize the temperature in the node numbered 60 (refer to the numbers given in Fig. 1) and keep temperature gradients small.		Here, <tt>c60</tt> refers to a specific output matrix <math>C</math>,
			which is defined to minimize the temperature in the node numbered 60 (refer to the numbers given in Fig. 1) and keep temperature gradients small.
	The latter task is taken into account by the inclusion of temperature differences between specific points in the interior and reference points on the boundary, e.g., temperature differences between nodes 83 and 34.		The latter task is taken into account by the inclusion of temperature differences between specific points in the interior and reference points on the boundary, e.g., temperature differences between nodes 83 and 34.
	Again refer to Fig. 1 for the nodes used.		Again refer to Fig. 1 for the nodes used.

Saak at 15:19, 4 October 2023

2023-10-04T15:19:58Z

@@ Line 8: / Line 8: @@
 [[Category:MIMO]]
 [[Category:Sparse]]
 ==Description: A Semi-discretized Heat Transfer Problem for Optimal Cooling of Steel Profiles==

Saak: /* Data */

2021-12-14T09:02:24Z

Data

← Older revision		Revision as of 09:02, 14 December 2021
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	\|max. mesh width		\|max. mesh width
	\|-		\|-
	\|[https://~~www~~.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e3-rail_1357.zip SteelProfile-dim1e3-rail_1357.zip] (95kB)		\|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e3-rail_1357.zip SteelProfile-dim1e3-rail_1357.zip] (95kB)
	\|<math>8\,985</math>		\|<math>8\,985</math>
	\|<math>8\,997</math>		\|<math>8\,997</math>
	\|<math>5.5280 \cdot 10^{-2}</math>		\|<math>5.5280 \cdot 10^{-2}</math>
	\|-		\|-
	\|[https://~~www~~.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e4-rail_5177.zip SteelProfile-dim1e4-rail_5177.zip] (299kB)		\|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e4-rail_5177.zip SteelProfile-dim1e4-rail_5177.zip] (299kB)
	\|<math>35\,185</math>		\|<math>35\,185</math>
	\|<math>35\,241</math>		\|<math>35\,241</math>
	\|<math>2.7640 \cdot 10^{-2}</math>		\|<math>2.7640 \cdot 10^{-2}</math>
	\|-		\|-
	\|[https://~~www~~.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e4-rail_20209.zip SteelProfile-dim1e4-rail_20209.zip] (1011kB)		\|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e4-rail_20209.zip SteelProfile-dim1e4-rail_20209.zip] (1011kB)
	\|<math>139\,233</math>		\|<math>139\,233</math>
	\|<math>139\,473</math>		\|<math>139\,473</math>
	\|<math>1.3820 \cdot 10^{-2}</math>		\|<math>1.3820 \cdot 10^{-2}</math>
	\|-		\|-
	\|[https://~~www~~.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e5-rail_79841.zip SteelProfile-dim1e5-rail_79841.zip] (3.7MB)		\|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SteelProfile-dim1e5-rail_79841.zip SteelProfile-dim1e5-rail_79841.zip] (3.7MB)
	\|<math>553\,921</math>		\|<math>553\,921</math>
	\|<math>554\,913</math>		\|<math>554\,913</math>

Himpe: Fix CrossRefs

2021-08-27T07:57:14Z

Fix CrossRefs

← Older revision		Revision as of 07:57, 27 August 2021
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	For the discretization we use the <tt>ALBERTA-1.2 fem-toolbox</tt> (see <ref name="SS00"/> for details).		For the discretization we use the <tt>ALBERTA-1.2 fem-toolbox</tt> (see <ref name="SS00"/> for details).
	We applied linear Lagrange elements and used a projection method for the curved boundaries.		We applied linear Lagrange elements and used a projection method for the curved boundaries.
	The initial mesh (see ~~<xr id="fig1"/>~~) was produced by MATLABs <tt>pdetool</tt>, which implements a [[wikipedia:Delaunay_triangulation\|Delaunay triangulation]] algorithm.		The initial mesh (see Fig. 1) was produced by MATLABs <tt>pdetool</tt>, which implements a [[wikipedia:Delaunay_triangulation\|Delaunay triangulation]] algorithm.
	The finer discretizations were produced by global mesh refinement using a bisection refinement method.		The finer discretizations were produced by global mesh refinement using a bisection refinement method.
	The discrete [[wikipedia:Linear–quadratic_regulator\|LQR]] problem is then: minimize [[#eq2\|(2)]] with respect to:		The discrete [[wikipedia:Linear–quadratic_regulator\|LQR]] problem is then: minimize [[#eq2\|(2)]] with respect to:
Line 134:		Line 134:

	The data sets are named <tt>rail_(problem dimension)_C60.(matrix name)</tt>.		The data sets are named <tt>rail_(problem dimension)_C60.(matrix name)</tt>.
	Here <tt>C60</tt> is a specific output matrix which is defined to minimize the temperature in the node numbered 60 (refer to the numbers given in ~~<xr id="fig1"/>~~) and keep temperature gradients small.		Here <tt>C60</tt> is a specific output matrix which is defined to minimize the temperature in the node numbered 60 (refer to the numbers given in Fig. 1) and keep temperature gradients small.
	The latter task is taken into account by the inclusion of temperature differences between specific points in the interior and reference points on the boundary, e.g., temperature differences between nodes 83 and 34.		The latter task is taken into account by the inclusion of temperature differences between specific points in the interior and reference points on the boundary, e.g., temperature differences between nodes 83 and 34.
	Again refer to ~~<xr id="fig1"/>~~ for the nodes used.		Again refer to Fig. 1 for the nodes used.
	The definitions of other output matrices that we tested can be found in <ref name="Saa03"/>.		The definitions of other output matrices that we tested can be found in <ref name="Saa03"/>.
	The problem resides at temperatures of approximately <math>1\,000</math> degrees centigrade down to about <math>500-700</math> degrees depending on calculation time.		The problem resides at temperatures of approximately <math>1\,000</math> degrees centigrade down to about <math>500-700</math> degrees depending on calculation time.

Werner at 07:56, 14 April 2020

2020-04-14T07:56:12Z

← Older revision		Revision as of 07:56, 14 April 2020
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	with <math>y=Cx</math> is minimized.		with <math>y=Cx</math> is minimized.
	Here, <math>Q</math> and <math>R</math> are linear self-adjoint operators on the output space <math>Y</math> and the control space <math>U</math> with <math>Q \geq 0,~ R > 0</math> and <math>C \in L(X,Y)</math>.		Here, <math>Q</math> and <math>R</math> are linear self-adjoint operators on the output space <math>Y</math> and the control space <math>U</math> with <math>Q \geq 0,~ R > 0</math> and <math>C \in L(X,Y)</math>.
	The variational formulation of [[#eq1\|(1)]] with <math>g_i(t,\chi) = q_i(u_i- x(\chi,t))</math> leads to:		The variational formulation of [[#eq1\|(1)]] with <math>g_i(t,\chi) = q_i(u_i- x(t,\chi))</math> leads to:

	:<math>		:<math>

Saak at 14:17, 10 March 2020

2020-03-10T14:17:34Z

← Older revision		Revision as of 14:17, 10 March 2020
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	Again refer to <xr id="fig1"/> for the nodes used.		Again refer to <xr id="fig1"/> for the nodes used.
	The definitions of other output matrices that we tested can be found in <ref name="Saa03"/>.		The definitions of other output matrices that we tested can be found in <ref name="Saa03"/>.
	The problem resides at temperatures of approximately <math>1\,000</math> degrees centigrade down to about <math>500--700</math> degrees depending on calculation time.		The problem resides at temperatures of approximately <math>1\,000</math> degrees centigrade down to about <math>500-700</math> degrees depending on calculation time.
	The state values are scaled to <math>1\,000</math> degrees centigrade being equivalent to <math>1.000</math>.		The state values are scaled to <math>1\,000</math> degrees centigrade being equivalent to <math>1.000</math>.
	This, together with the scaling of the domain, results in a scaling of the time line with factor <math>100</math>, meaning that calculated times have to be divided by <math>100</math> to get the real time in seconds.		This, together with the scaling of the domain, results in a scaling of the time line with factor <math>100</math>, meaning that calculated times have to be divided by <math>100</math> to get the real time in seconds.

Saak: /* Dimensions */

2019-11-17T14:08:21Z

Dimensions

← Older revision		Revision as of 14:08, 17 November 2019
Line 163:		Line 163:
	System variants:		System variants:

	<tt>~~rail1375~~</tt>: <math>N = 1\,~~375~~</math>,		<tt>rail1357</tt>: <math>N = 1\,357</math>,
	<tt>rail5177</tt>: <math>N = 5\,177</math>,		<tt>rail5177</tt>: <math>N = 5\,177</math>,
	<tt>rail20209</tt>: <math>N = 20\,209</math>,		<tt>rail20209</tt>: <math>N = 20\,209</math>,

Saak at 14:07, 31 July 2018

2018-07-31T14:07:15Z

← Older revision		Revision as of 14:07, 31 July 2018
Line 41:		Line 41:
	</math>		</math>

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

Himpe: minor updates.

2018-07-30T10:22:59Z

minor updates.

← Older revision		Revision as of 10:22, 30 July 2018
Line 20:		Line 20:

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

	The scientific challenge here is to give the engineers a tool to precalculate different control laws yielding different temperature distributions in order to decide which cooling strategy to choose.		The scientific challenge here is to give the engineers a tool to precalculate different control laws yielding different temperature distributions in order to decide which cooling strategy to choose.
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	We can only briefly introduce the model here for details we refer to <ref name="Saa03"/>, <ref name="BenS05b"/>, or <ref name="bs04"/>.		We can only briefly introduce the model here for details we refer to <ref name="Saa03"/>, <ref name="BenS05b"/>, or <ref name="bs04"/>.
	We assume an infinitely long steel profile so that we may restrict ourselves to a 2D model.		We assume an infinitely long steel profile so that we may restrict ourselves to a 2D model.
	Exploiting the symmetry of the workpiece, the computational domain <math>\Omega \subset \mathbb{R}^2</math> is chosen as ~~the~~ half of a cross section of the rail profile.		Exploiting the symmetry of the workpiece, the computational domain <math>\Omega \subset \mathbb{R}^2</math> is chosen as half a cross section of the rail profile.
	The heat distribution is modeled by the unsteady linear heat equation on <math>\Omega</math>:		The heat distribution is modeled by the unsteady linear heat equation on <math>\Omega</math>:

Line 41:		Line 41:
	</math>		</math>

	where <math>x</math> is the temperature distribution <math>(\in H^1([0,\infty],X)</math> with <math>X:=H^1(\Omega)</math> the state space), <math>c</math> the specific heat capacity, <math>\lambda</math> the heat conductivity and <math>\rho</math> the density of the rail profile.		where <math>x</math> is the temperature distribution (<math>x \in H^1([0,\infty],X)</math> with <math>X:=H^1(\Omega)</math> the state space), <math>c</math> the specific heat capacity, <math>\lambda</math> the heat conductivity and <math>\rho</math> the density of the rail profile.
	We split the boundary into several parts <math>\Gamma_i</math> on which we have different boundary functions <math>g_i</math>,		We split the boundary into several parts <math>\Gamma_i</math> on which we have different boundary functions <math>g_i</math>,
	allowing us to vary controls on different parts of the surface.		allowing us to vary controls on different parts of the surface.
Line 138:		Line 138:
	Again refer to <xr id="fig1"/> for the nodes used.		Again refer to <xr id="fig1"/> for the nodes used.
	The definitions of other output matrices that we tested can be found in <ref name="Saa03"/>.		The definitions of other output matrices that we tested can be found in <ref name="Saa03"/>.
	The problem resides at temperatures of approximately <math>1\,000</math> degrees centigrade down to about <math>500-700</math> degrees depending on calculation time.		The problem resides at temperatures of approximately <math>1\,000</math> degrees centigrade down to about <math>500--700</math> degrees depending on calculation time.
	The state values are scaled to <math>1\,000</math> degrees centigrade being equivalent to <math>1.000</math>.		The state values are scaled to <math>1\,000</math> degrees centigrade being equivalent to <math>1.000</math>.
	This, together with the scaling of the domain, results in a scaling of the time line with factor <math>100</math>, meaning that calculated times have to be divided by <math>100</math> to get the real time in seconds.		This, together with the scaling of the domain, results in a scaling of the time line with factor <math>100</math>, meaning that calculated times have to be divided by <math>100</math> to get the real time in seconds.
Line 185:		Line 185:

	@TechReport{BenS05b,		@TechReport{BenS05b,
	title = {Linear-Quadratic Regulator Design for Optimal Cooling of Steel		title = {Linear-Quadratic Regulator Design for Optimal Cooling of Steel Profiles},
	Profiles},
	author = {P. Benner and J. Saak},		author = {P. Benner and J. Saak},
	institution = {Sonderforschungsbereich 393 {\itshape Parallele Numerische		institution = {Sonderforschungsbereich 393 {\itshape Parallele Numerische Simulation f\"ur Physik und Kontinuumsmechanik}, TU Chem\-nitz},
	Simulation f\"ur Physik und Kontinuumsmechanik}, TU
	Chem\-nitz},
	~~year = 2005~~,
	address = {D-09107 Chem\-nitz (Germany)},		address = {D-09107 Chem\-nitz (Germany)},
	number = {SFB393/05-05},		number = {SFB393/05-05},
			year = {2005},
	url = {<nowiki>http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601597</nowiki>}		url = {<nowiki>http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601597</nowiki>}
	}		}