Anemometer - Revision history

Schuetz2 at 08:42, 11 June 2025

2025-06-11T08:42:50Z

← Older revision		Revision as of 08:42, 11 June 2025
Line 172:		Line 172:
	title = {Anemometer},		title = {Anemometer},
	howpublished = {{MORwiki} -- Model Order Reduction Wiki},		howpublished = {{MORwiki} -- Model Order Reduction Wiki},
	url = <nowiki>{~~http~~://modelreduction.org/index.php/Anemometer}</nowiki>,		url = <nowiki>{https://modelreduction.org/morwiki/index.php/Anemometer}</nowiki>,
	year = {2018}		year = {2018}
	}		}

Malhotra at 09:24, 30 November 2023

2023-11-30T09:24:16Z

@@ Line 8: / Line 8: @@
 [[Category:first differential order]]
 [[Category:SISO]]
 ==Description==

Mlinaric: Math fixes

2023-08-29T01:22:37Z

Math fixes

← Older revision		Revision as of 01:22, 29 August 2023
Line 29:		Line 29:
	The physical model can be expressed by the [[wikipedia:Convection–diffusion_equation\|convection-diffusion partial differential equation]] <ref name="moosmann04" group="b)"/>:		The physical model can be expressed by the [[wikipedia:Convection–diffusion_equation\|convection-diffusion partial differential equation]] <ref name="moosmann04" group="b)"/>:

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

	where <math>\rho</math> denotes the mass density, <math>c</math> is the specific heat capacity, <math>\kappa</math> is the thermal conductivity,		where <math>\rho</math> denotes the mass density, <math>c</math> is the specific heat capacity, <math>\kappa</math> is the thermal conductivity,
Line 43:		Line 47:
	The <math>n</math> dimensional [[wikipedia:Ordinary_Differential_Equation\|ODE]] system has the following transfer function		The <math>n</math> dimensional [[wikipedia:Ordinary_Differential_Equation\|ODE]] system has the following transfer function
	:<math>		:<math>
	G(p) = C(~~(sE~~ - A_1- p(A_2 - A_1))^{-1}B)		G(s, p) = C (s E - A_1 - p (A_2 - A_1))^{-1} B
	</math>		</math>

Line 56:		Line 60:

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

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

	==Origin==		==Origin==
Line 95:		Line 105:

	:<math>		:<math>
	\begin{~~array}{rcl~~}		\begin{align}
	E\dot{x}(t) &=& (A_1 + p (A_2 - A_1))x(t) + B \\		E \dot{x}(t) &= (A_1 + p (A_2 - A_1)) x(t) + B u(t) \\
	y(t) &=& Cx(t)		y(t) &= Cx(t)
	\end{~~array~~}		\end{align}
	</math>		</math>

Line 112:		Line 122:

	:<math>		:<math>
	\begin{~~array}{rcl~~}		\begin{align}
	(E_1 + p_0 (E_2 - E_1))\dot{x}(t) &=& (A_1 + p_1 (A_3 - A_1 + A_4 - A_5) + p_2 (A_2 - A_1))x(t) + B \\		(E_1 + p_0 (E_2 - E_1)) \dot{x}(t) &= (A_1 + p_1 (A_3 - A_1 + A_4 - A_5) + p_2 (A_2 - A_1)) x(t) + B u(t) \\
	y(t) &=& Cx(t)		y(t) &= Cx(t)
	\end{~~array~~}		\end{align}
	</math>		</math>

Lund: /* Dimensions */ simplify matrix indexing

2023-05-02T13:52:01Z

Dimensions: simplify matrix indexing

← Older revision		Revision as of 13:52, 2 May 2023
Line 104:		Line 104:

	<math>E \in \mathbb{R}^{29008 \times 29008}</math>,		<math>E \in \mathbb{R}^{29008 \times 29008}</math>,
	<math>~~A_1 \in \mathbb~~{~~R}^{29008 \times 29008~~}~~</math>,~~		<math>A_{1,2} \in \mathbb{R}^{29008 \times 29008}</math>,
	~~<math>A_2~~ \in \mathbb{R}^{29008 \times 29008}</math>,
	<math>B \in \mathbb{R}^{29008 \times 1}</math>,		<math>B \in \mathbb{R}^{29008 \times 1}</math>,
	<math>C \in \mathbb{R}^{1 \times 29008}</math>.		<math>C \in \mathbb{R}^{1 \times 29008}</math>.
Line 121:		Line 120:
	System dimensions:		System dimensions:

	<math>~~E_1 \in \mathbb~~{~~R}^{29008 \times 29008~~}~~</math>,~~		<math>E_{1,2} \in \mathbb{R}^{29008 \times 29008}</math>,
	~~<math>E_2~~ \in \mathbb{R}^{29008 \times 29008}</math>,		<math>A_{1,2,3,4,5} \in \mathbb{R}^{29008 \times 29008}</math>,
	<math>~~A_1 \in \mathbb~~{~~R}^{29008 \times 29008}</math>~~,
	~~<math>A_2 \in \mathbb{R}^{29008 \times 29008}</math>~~,
	~~<math>A_3 \in \mathbb{R}^{29008 \times 29008}</math>~~,
	~~<math>A_4 \in \mathbb{R~~}~~^{29008 \times 29008}</math>,~~
	~~<math>A_5~~ \in \mathbb{R}^{29008 \times 29008}</math>,
	<math>B \in \mathbb{R}^{29008 \times 1}</math>,		<math>B \in \mathbb{R}^{29008 \times 1}</math>,
	<math>C \in \mathbb{R}^{1 \times 29008}</math>.		<math>C \in \mathbb{R}^{1 \times 29008}</math>.


	==Citation==		==Citation==

Lund: b --> B

2022-09-01T14:07:21Z

b --> B

← Older revision		Revision as of 14:07, 1 September 2022
Line 43:		Line 43:
	The <math>n</math> dimensional [[wikipedia:Ordinary_Differential_Equation\|ODE]] system has the following transfer function		The <math>n</math> dimensional [[wikipedia:Ordinary_Differential_Equation\|ODE]] system has the following transfer function
	:<math>		:<math>
	G(p) = C((sE - A_1- p(A_2 - A_1))^{-1}b)		G(p) = C((sE - A_1- p(A_2 - A_1))^{-1}B)
	</math>		</math>

	with the fluid velocity <math>p(=v)</math> as single parameter.		with the fluid velocity <math>p(=v)</math> as single parameter.
	Here <math>E</math> is the heat capacitance matrix, <math>b</math> is the load vector which is derived from separating the spatial and temporal variables in <math>\dot{q}</math> and the [[wikipedia:Finite_Element_Method\|FEM]] discretization w.r.t. the spatial variables.		Here <math>E</math> is the heat capacitance matrix, <math>B</math> is the load vector which is derived from separating the spatial and temporal variables in <math>\dot{q}</math> and the [[wikipedia:Finite_Element_Method\|FEM]] discretization w.r.t. the spatial variables.
	<math>A_i</math> are the stiffness matrices with <math>i=1</math> for pure diffusion and <math>i=2</math> for diffusion and convection.		<math>A_i</math> are the stiffness matrices with <math>i=1</math> for pure diffusion and <math>i=2</math> for diffusion and convection.
	Thus, for obtaining pure convection you have to compute <math>A_2 - A_1</math>.		Thus, for obtaining pure convection you have to compute <math>A_2 - A_1</math>.
Line 56:		Line 56:

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

Line 96:		Line 96:
	:<math>		:<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	E\dot{x}(t) &=& (A_1 + p (A_2 - A_1))x(t) + b \\		E\dot{x}(t) &=& (A_1 + p (A_2 - A_1))x(t) + B \\
	y(t) &=& Cx(t)		y(t) &=& Cx(t)
	\end{array}		\end{array}
Line 106:		Line 106:
	<math>A_1 \in \mathbb{R}^{29008 \times 29008}</math>,		<math>A_1 \in \mathbb{R}^{29008 \times 29008}</math>,
	<math>A_2 \in \mathbb{R}^{29008 \times 29008}</math>,		<math>A_2 \in \mathbb{R}^{29008 \times 29008}</math>,
	<math>b \in \mathbb{R}^{29008 \times 1}</math>,		<math>B \in \mathbb{R}^{29008 \times 1}</math>,
	<math>C \in \mathbb{R}^{1 \times 29008}</math>.		<math>C \in \mathbb{R}^{1 \times 29008}</math>.

Line 114:		Line 114:
	:<math>		:<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	(E_1 + p_0 (E_2 - E_1))\dot{x}(t) &=& (A_1 + p_1 (A_3 - A_1 + A_4 - A_5) + p_2 (A_2 - A_1))x(t) + b \\		(E_1 + p_0 (E_2 - E_1))\dot{x}(t) &=& (A_1 + p_1 (A_3 - A_1 + A_4 - A_5) + p_2 (A_2 - A_1))x(t) + B \\
	y(t) &=& Cx(t)		y(t) &=& Cx(t)
	\end{array}		\end{array}
Line 128:		Line 128:
	<math>A_4 \in \mathbb{R}^{29008 \times 29008}</math>,		<math>A_4 \in \mathbb{R}^{29008 \times 29008}</math>,
	<math>A_5 \in \mathbb{R}^{29008 \times 29008}</math>,		<math>A_5 \in \mathbb{R}^{29008 \times 29008}</math>,
	<math>b \in \mathbb{R}^{29008 \times 1}</math>,		<math>B \in \mathbb{R}^{29008 \times 1}</math>,
	<math>C \in \mathbb{R}^{1 \times 29008}</math>.		<math>C \in \mathbb{R}^{1 \times 29008}</math>.

Himpe at 12:59, 25 August 2022

2022-08-25T12:59:15Z

← Older revision		Revision as of 12:59, 25 August 2022
Line 4:		Line 4:
	[[Category:ODE]]		[[Category:ODE]]
	[[Category:time invariant]]		[[Category:time invariant]]
	[[Category:~~parametric 1 parameter]]~~		[[Category:Parametric]]
	~~[[Category:parametric 2-5 parameters~~]]
	[[Category:affine parameter representation]]		[[Category:affine parameter representation]]
	[[Category:first differential order]]		[[Category:first differential order]]

Himpe: load vector fixes

2022-04-23T15:56:56Z

load vector fixes

← Older revision		Revision as of 15:56, 23 April 2022
Line 44:		Line 44:
	The <math>n</math> dimensional [[wikipedia:Ordinary_Differential_Equation\|ODE]] system has the following transfer function		The <math>n</math> dimensional [[wikipedia:Ordinary_Differential_Equation\|ODE]] system has the following transfer function
	:<math>		:<math>
	G(p) = C((sE - A_1- p(A_2 - A_1))^{-1}B)		G(p) = C((sE - A_1- p(A_2 - A_1))^{-1}b)
	</math>		</math>

	with the fluid velocity <math>p(=v)</math> as single parameter.		with the fluid velocity <math>p(=v)</math> as single parameter.
	Here <math>E</math> is the heat capacitance matrix, <math>B</math> is the load vector which is derived from separating the spatial and temporal variables in <math>\dot{q}</math> and the [[wikipedia:Finite_Element_Method\|FEM]] discretization w.r.t. the spatial variables.		Here <math>E</math> is the heat capacitance matrix, <math>b</math> is the load vector which is derived from separating the spatial and temporal variables in <math>\dot{q}</math> and the [[wikipedia:Finite_Element_Method\|FEM]] discretization w.r.t. the spatial variables.
	<math>A_i</math> are the stiffness matrices with <math>i=1</math> for pure diffusion and <math>i=2</math> for diffusion and convection.		<math>A_i</math> are the stiffness matrices with <math>i=1</math> for pure diffusion and <math>i=2</math> for diffusion and convection.
	Thus, for obtaining pure convection you have to compute <math>A_2 - A_1</math>.		Thus, for obtaining pure convection you have to compute <math>A_2 - A_1</math>.
Line 57:		Line 57:

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

Line 97:		Line 97:
	:<math>		:<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	E\dot{x}(t) &=& (A_1 + p (A_2 - A_1))x(t) + ~~Bu(t)~~ \\		E\dot{x}(t) &=& (A_1 + p (A_2 - A_1))x(t) + b \\
	y(t) &=& Cx(t)		y(t) &=& Cx(t)
	\end{array}		\end{array}
Line 107:		Line 107:
	<math>A_1 \in \mathbb{R}^{29008 \times 29008}</math>,		<math>A_1 \in \mathbb{R}^{29008 \times 29008}</math>,
	<math>A_2 \in \mathbb{R}^{29008 \times 29008}</math>,		<math>A_2 \in \mathbb{R}^{29008 \times 29008}</math>,
	<math>B \in \mathbb{R}^{29008 \times 1}</math>,		<math>b \in \mathbb{R}^{29008 \times 1}</math>,
	<math>C \in \mathbb{R}^{1 \times 29008}</math>.		<math>C \in \mathbb{R}^{1 \times 29008}</math>.

Line 115:		Line 115:
	:<math>		:<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	(E_1 + p_0 (E_2 - E_1))\dot{x}(t) &=& (A_1 + p_1 (A_3 - A_1 + A_4 - A_5) + p_2 (A_2 - A_1))x(t) + ~~Bu(t)~~ \\		(E_1 + p_0 (E_2 - E_1))\dot{x}(t) &=& (A_1 + p_1 (A_3 - A_1 + A_4 - A_5) + p_2 (A_2 - A_1))x(t) + b \\
	y(t) &=& Cx(t)		y(t) &=& Cx(t)
	\end{array}		\end{array}
Line 129:		Line 129:
	<math>A_4 \in \mathbb{R}^{29008 \times 29008}</math>,		<math>A_4 \in \mathbb{R}^{29008 \times 29008}</math>,
	<math>A_5 \in \mathbb{R}^{29008 \times 29008}</math>,		<math>A_5 \in \mathbb{R}^{29008 \times 29008}</math>,
	<math>B \in \mathbb{R}^{29008 \times 1}</math>,		<math>b \in \mathbb{R}^{29008 \times 1}</math>,
	<math>C \in \mathbb{R}^{1 \times 29008}</math>.		<math>C \in \mathbb{R}^{1 \times 29008}</math>.

Himpe: Fix CrossRefs

2021-08-27T07:58:52Z

Fix CrossRefs

← Older revision		Revision as of 07:58, 27 August 2021
Line 22:		Line 22:

	An '''Anemometer'''<ref name="ernst01" group="a)"/><ref name="benner05" group="a)"/><ref name="moosmann05" group="a)"/><ref name="moosmann07" group="c)"/><ref name="moosmann05b" group="c)"/><ref name="rudnyi06" group="c)"/> (see [[wikipedia:Thermal_mass_flow_meter\|thermal mass flow meter]])		An '''Anemometer'''<ref name="ernst01" group="a)"/><ref name="benner05" group="a)"/><ref name="moosmann05" group="a)"/><ref name="moosmann07" group="c)"/><ref name="moosmann05b" group="c)"/><ref name="rudnyi06" group="c)"/> (see [[wikipedia:Thermal_mass_flow_meter\|thermal mass flow meter]])
	is a flow sensing device, consisting of a heater and temperature sensors before and after the heater, placed either directly in the flow or in its vicinity ~~(see <xr id="fig:plot1"/>)~~.		is a flow sensing device, consisting of a heater and temperature sensors before and after the heater, placed either directly in the flow or in its vicinity Fig. 1.
	They are located on a membrane to minimize heat dissipation through the structure.		They are located on a membrane to minimize heat dissipation through the structure.
	Without any flow, the heat dissipates symmetrically into the fluid.		Without any flow, the heat dissipates symmetrically into the fluid.
	This symmetry is disturbed if a flow is applied to the fluid,		This symmetry is disturbed if a flow is applied to the fluid,
	which leads to a convection on the temperature field and therefore to a difference between the temperature sensors (see ~~<xr id="fig:plot2"/>~~) from which the fluid velocity can be determined.		which leads to a convection on the temperature field and therefore to a difference between the temperature sensors (see Fig. 2) from which the fluid velocity can be determined.

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

Himpe: Add dimensions section and various other fixes.

2018-11-19T11:30:54Z

Add dimensions section and various other fixes.

@@ Line 21: / Line 21: @@
 </figure>
-An '''Anemometer'''<ref name="ernst01" group="a)"/><ref name="benner05" group="a)"/><ref name="moosmann05" group="a)"/><ref name="moosmann07" group="c)"/><ref name="moosmann05b" group="c)"/><ref name="rudnyi06" group="c)"/>
+An '''Anemometer'''<ref name="ernst01" group="a)"/><ref name="benner05" group="a)"/><ref name="moosmann05" group="a)"/><ref name="moosmann07" group="c)"/><ref name="moosmann05b" group="c)"/><ref name="rudnyi06" group="c)"/> (see [[wikipedia:Thermal_mass_flow_meter|thermal mass flow meter]])
 is a flow sensing device, consisting of a heater and temperature sensors before and after the heater, placed either directly in the flow or in its vicinity (see <xr id="fig:plot1"/>).
 They are located on a membrane to minimize heat dissipation through the structure.
@@ Line 88: / Line 88: @@
 To test the quality of the reduced order systems, harmonic simulations as well as transient step responses could be computed, see <ref name="baur11" group="c)"/>.
 ==Citation==
@@ Line 101: / Line 145: @@
     howpublished = {{MORwiki} -- Model Order Reduction Wiki},
     url =          <nowiki>{http://modelreduction.org/index.php/Anemometer}</nowiki>,
-    year =         {20XX}
+    year =         {2018}
+  }

Werner: Updated BibTeX entry.

2018-04-24T08:47:04Z

Updated BibTeX entry.

← Older revision		Revision as of 08:47, 24 April 2018
Line 94:		Line 94:

	* For the benchmark itself and its data:		* For the benchmark itself and its data:
	::The MORwiki Community. '''Anemometer'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Anemometer		::The MORwiki Community, '''Anemometer'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Anemometer
	@MISC{morwiki_anemom,
	author = {The {MORwiki} ~~Community~~},		@MISC{morwiki_anemom,
	title = {Anemometer},		author = <nowiki>{{The MORwiki Community}}</nowiki>,
	howpublished = {{MORwiki} -- Model Order Reduction Wiki},		title = {Anemometer},
	url = {<nowiki>http://modelreduction.org/index.php/Anemometer</nowiki>},		howpublished = {{MORwiki} -- Model Order Reduction Wiki},
	year = {20XX}		url = <nowiki>{http://modelreduction.org/index.php/Anemometer}</nowiki>,
	}		year = {20XX}
			}

	* For the background on the benchmark:		* For the background on the benchmark: