Electrostatic Beam - Revision history

Schuetz2 at 05:28, 12 June 2025

2025-06-12T05:28:27Z

← Older revision		Revision as of 05:28, 12 June 2025
Line 196:		Line 196:
	title = {Electrostatic Beam},		title = {Electrostatic Beam},
	howpublished = {{MORwiki} -- Model Order Reduction Wiki},		howpublished = {{MORwiki} -- Model Order Reduction Wiki},
	url = <nowiki>{~~http~~://modelreduction.org/~~index.php~~/Electrostatic_Beam}</nowiki>,		url = <nowiki>{https://modelreduction.org/morwiki/Electrostatic_Beam}</nowiki>,
	year = {20XX}		year = {20XX}
	}		}

Malhotra: infobox string

2023-11-30T09:31:12Z

infobox string

@@ Line 1: / Line 1: @@
 [[Category:benchmark]]
 [[Category:Oberwolfach]]
 ==Description: Beam Actuated by Electrostatic Force==

Mitchell: typo

2022-03-14T15:54:06Z

typo

← Older revision		Revision as of 15:54, 14 March 2022
Line 132:		Line 132:
	</math>		</math>

	where <math>B</math> is a <math>n \times 1</math> ~~matrices~~ with <math>1</math> at all charge DOFs of the upper beam and <math>C</math> is a <math>1 \times n</math> matrix with the only nonzero entry at the <math>y</math> DOF of the middle node.		where <math>B</math> is a <math>n \times 1</math> matrix with <math>1</math> at all charge DOFs of the upper beam and <math>C</math> is a <math>1 \times n</math> matrix with the only nonzero entry at the <math>y</math> DOF of the middle node.

	Details of the implementation are available in a separate report ([https://www.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ElectrostaticBeam.pdf ElectrostaticBeam.pdf]), see also <ref name="lienemann2006"/>.		Details of the implementation are available in a separate report ([https://www.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ElectrostaticBeam.pdf ElectrostaticBeam.pdf]), see also <ref name="lienemann2006"/>.

Mitchell: grammar

2022-03-14T15:52:11Z

grammar

← Older revision		Revision as of 15:52, 14 March 2022
Line 17:		Line 17:

	Accurate modelling of such a system typically leads to high order models.		Accurate modelling of such a system typically leads to high order models.
	The tasks of simulation, analysis and controller design of high order nonlinear control systems can be simplified by reducing the order of the original system and ~~approximate~~ it by a lower order model.		The tasks of simulation, analysis and controller design of high order nonlinear control systems can be simplified by reducing the order of the original system and approximating it by a lower order model.

	An application of electrostatic moving structures are e.g. [[wikipedia:RF_switch\|RF switches]] or filters.		An application of electrostatic moving structures are e.g. [[wikipedia:RF_switch\|RF switches]] or filters.

Saak: /* Data */

2021-12-14T08:54:49Z

Data

← Older revision		Revision as of 08:54, 14 December 2021
Line 112:		Line 112:
	! Compressed Size [kB]		! Compressed Size [kB]
	\|-		\|-
	\|[https://~~www~~.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ElectrostaticBeam-dim1e1-E10.zip ElectrostaticBeam-dim1e1-E10.zip]		\|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ElectrostaticBeam-dim1e1-E10.zip ElectrostaticBeam-dim1e1-E10.zip]
	\|10		\|10
	\|38		\|38
	\|4144		\|4144
	\|-		\|-
	\|[https://~~www~~.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ElectrostaticBeam-dim1e2-E100.zip ElectrostaticBeam-dim1e2-E100.zip]		\|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ElectrostaticBeam-dim1e2-E100.zip ElectrostaticBeam-dim1e2-E100.zip]
	\|100		\|100
	\|398		\|398

Himpe: Fix CrossRefs

2021-08-27T07:32:31Z

Fix CrossRefs

← Older revision		Revision as of 07:32, 27 August 2021
Line 24:		Line 24:
	===Model description===		===Model description===

	This model describes a slender beam which is actuated by a voltage between the beam and the ground electrode below (see ~~<xr id="fig1"/>~~).		This model describes a slender beam which is actuated by a voltage between the beam and the ground electrode below (see Fig. 1).

	On the beam, at least three degrees of freedom per node have to be considered:		On the beam, at least three degrees of freedom per node have to be considered:

Himpe: verified electrostatic beam

2018-07-24T07:58:51Z

verified electrostatic beam

← Older revision		Revision as of 07:58, 24 July 2018
Line 1:		Line 1:
	~~{{preliminary}} <!-- Do not remove -->~~


	[[Category:benchmark]]		[[Category:benchmark]]
	[[Category:Oberwolfach]]		[[Category:Oberwolfach]]
Line 12:		Line 9:
	Moving structures are an essential part for many microsystem devices, among them fluidic components like pumps and electrically controllable valves, sensing cantilevers, and optical structures.		Moving structures are an essential part for many microsystem devices, among them fluidic components like pumps and electrically controllable valves, sensing cantilevers, and optical structures.

	Several actuation principles can be employed on microscopic length scales, ~~the~~ most frequent ~~certainly the~~ electromagnetic forces.		Several actuation principles can be employed on microscopic length scales, certainly most frequent electromagnetic forces.
	While electrostatic actuation falls behind at the macro scale, the effect of charged bodies outperforms magnetic forces in the micro scale both in terms of performance and ~~fabrication~~ expense.		While electrostatic actuation falls behind at the macro scale, the effect of charged bodies outperforms magnetic forces in the micro scale both in terms of performance and manufacturing expense.

	While the single component can easily be simulated on a usual desktop computer, the calculation of a system of many coupled devices still presents a challenge.		While the single component can easily be simulated on a usual desktop computer, the calculation of a system of many coupled devices still presents a challenge.
Line 22:		Line 19:
	The tasks of simulation, analysis and controller design of high order nonlinear control systems can be simplified by reducing the order of the original system and approximate it by a lower order model.		The tasks of simulation, analysis and controller design of high order nonlinear control systems can be simplified by reducing the order of the original system and approximate it by a lower order model.

	A application of electrostatic moving structures are e.g. RF switches or filters.		An application of electrostatic moving structures are e.g. [[wikipedia:RF_switch\|RF switches]] or filters.
	Given a simple enough shape, they often can be modelled as one-dimensional beams embedded in two or three dimensional space.		Given a simple enough shape, they often can be modelled as one-dimensional beams embedded in two or three dimensional space.

Line 49:		Line 46:

	The calculation of the electrostatic force would require a boundary element discretization, where it would be necessary to recalculate the capacity matrix for each time-step due to the motion of the charges		The calculation of the electrostatic force would require a boundary element discretization, where it would be necessary to recalculate the capacity matrix for each time-step due to the motion of the charges
	This would require an integration over the beam's elements.		This would require an integration over the beam's elements and could be written in analytical form by using e.g. [[wikipedia:Gaussian_quadrature\|Gauss integration]];
	~~This~~ could be written in analytical form by using e.g. Gauss integration;
	however, the complexity of the resulting system would be too high.		however, the complexity of the resulting system would be too high.
	We therefore use the method shown in <ref name="siverberg1996"/>, i.e. we concentrate the charges on the nodes.		We therefore use the method shown in <ref name="siverberg1996"/>, i.e. we concentrate the charges on the nodes.
Line 79:		Line 75:
	\|-		\|-
	\|Beam width (w)		\|Beam width (w)
	\|<math>15 10^{-6} m</math>		\|<math>15 \cdot 10^{-6} m</math>
	\|-		\|-
	\|Distance between beams (s)		\|Distance between beams (s)
	\|<math>200 10^{-9} m</math>		\|<math>200 \cdot 10^{-9} m</math>
	\|-		\|-
	\|Material density (rho)		\|Material density (rho)
Line 88:		Line 84:
	\|-		\|-
	\|Cross-sectional area (A)		\|Cross-sectional area (A)
	\|<math>~~150e-~~12 ~~m12~~</math>		\|<math>150 \cdot 10^{12} m^2</math>
	\|-		\|-
	\|Moment of inertia (I)		\|Moment of inertia (I)
	\|<math>1.~~25e~~-21 m^4</math>		\|<math>1.25 \cdot 10^{-21} m^4</math>
	\|-		\|-
	\|Modulus of elasticity (E)		\|Modulus of elasticity (E)
	\|<math>1.~~31e11~~ Pa</math>		\|<math>1.31 \cdot 10^{11} Pa</math>
	\|-		\|-
	\|Contribution of <math>M</math> to damping		\|Contribution of <math>M</math> to damping
Line 167:		Line 163:
	<tt>E10</tt>: <math>N = 38</math>, <math>S = 28</math>,		<tt>E10</tt>: <math>N = 38</math>, <math>S = 28</math>,
	<tt>E100</tt>: <math>N = 398</math>, <math>S = 298</math>.		<tt>E100</tt>: <math>N = 398</math>, <math>S = 298</math>.

			==Citation==

			To cite this benchmark, use the following references:

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

			@MISC{morwiki_ebeam,
			author = <nowiki>{{The MORwiki Community}}</nowiki>,
			title = {Electrostatic Beam},
			howpublished = {{MORwiki} -- Model Order Reduction Wiki},
			url = <nowiki>{http://modelreduction.org/index.php/Electrostatic_Beam}</nowiki>,
			year = {20XX}
			}

			* For the background on the benchmark:

			@ARTICLE{morLieRK06,
			author = <nowiki>{J. Lienemann and E.B. Rudnyi and J.G. Korvink}</nowiki>,
			title = {{MST} {MEMS} model order reduction: Requirements and benchmarks},
			journal = {Journal of Biomechanics},
			volume = {415},
			number = {2--3},
			pages = {469--498},
			year = {2006},
			doi = {10.1016/j.laa.2005.04.002}
			}

	==References==		==References==

Himpe: /* Data */ Update benchmark data file links

2018-07-17T10:21:10Z

Data: Update benchmark data file links

← Older revision		Revision as of 10:21, 17 July 2018
Line 116:		Line 116:
	! Compressed Size [kB]		! Compressed Size [kB]
	\|-		\|-
	\|[https://~~portal~~.~~uni~~-~~freiburg~~.de/~~imteksimulation~~/~~downloads~~/~~benchmark~~/~~Eletrostatic%20beam%20%2838882%29/files/fileinnercontentproxy~~.~~2010~~-02-~~08.3711017358~~ E10.zip]		\|[https://www.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ElectrostaticBeam-dim1e1-E10.zip ElectrostaticBeam-dim1e1-E10.zip]
	\|10		\|10
	\|38		\|38
	\|4144		\|4144
	\|-		\|-
	\|[https://~~portal~~.~~uni~~-~~freiburg~~.de/~~imteksimulation~~/~~downloads~~/~~benchmark~~/~~Eletrostatic%20beam%20%2838882%29/files/fileinnercontentproxy~~.~~2010~~-02-~~08.3729192050~~ E100.zip]		\|[https://www.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ElectrostaticBeam-dim1e2-E100.zip ElectrostaticBeam-dim1e2-E100.zip]
	\|100		\|100
	\|398		\|398
Line 138:		Line 138:
	where <math>B</math> is a <math>n \times 1</math> matrices with <math>1</math> at all charge DOFs of the upper beam and <math>C</math> is a <math>1 \times n</math> matrix with the only nonzero entry at the <math>y</math> DOF of the middle node.		where <math>B</math> is a <math>n \times 1</math> matrices with <math>1</math> at all charge DOFs of the upper beam and <math>C</math> is a <math>1 \times n</math> matrix with the only nonzero entry at the <math>y</math> DOF of the middle node.

	Details of the implementation are available in a separate report ([https://~~portal~~.~~uni~~-~~freiburg~~.de/~~imteksimulation/downloads/benchmark~~/~~Eletrostatic%20beam%20%2838882%29~~/~~files~~/~~fileinnercontentproxy.2010-02-08~~.~~3748527563 ElectrostatBeam~~.pdf]), see also <ref name="lienemann2006"/>.		Details of the implementation are available in a separate report ([https://www.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ElectrostaticBeam.pdf ElectrostaticBeam.pdf]), see also <ref name="lienemann2006"/>.

	A typical input to this system is a step response; periodic on/off switching is also possible.		A typical input to this system is a step response; periodic on/off switching is also possible.

Saak at 14:48, 6 March 2018

2018-03-06T14:48:35Z

← Older revision		Revision as of 14:48, 6 March 2018
Line 3:		Line 3:

	[[Category:benchmark]]		[[Category:benchmark]]
			[[Category:Oberwolfach]]


	==Description: Beam Actuated by Electrostatic Force==		==Description: Beam Actuated by Electrostatic Force==

Himpe: Added Electrostatic Beam

2018-03-01T12:44:20Z

Added Electrostatic Beam

New page

{{preliminary}} 

[[Category:benchmark]]

==Description: Beam Actuated by Electrostatic Force==

<figure id="fig1">[[File:EBeam.gif|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Beam4.gif|490px|thumb|right|<caption>Degrees of freedom <math>y</math> and <math>\theta_z</math></caption>]]</figure>

Moving structures are an essential part for many microsystem devices, among them fluidic components like pumps and electrically controllable valves, sensing cantilevers, and optical structures.

Several actuation principles can be employed on microscopic length scales, the most frequent certainly the electromagnetic forces.
While electrostatic actuation falls behind at the macro scale, the effect of charged bodies outperforms magnetic forces in the micro scale both in terms of performance and fabrication expense.

While the single component can easily be simulated on a usual desktop computer, the calculation of a system of many coupled devices still presents a challenge.
This challenge is raised by the fact that many of these devices show a nonlinear behavior.
Especially for electrostatic structures, a further difficulty is the large reach of the electrostatic forces, leading to a strong spatial coupling of charges.

Accurate modelling of such a system typically leads to high order models.
The tasks of simulation, analysis and controller design of high order nonlinear control systems can be simplified by reducing the order of the original system and approximate it by a lower order model.

A application of electrostatic moving structures are e.g. RF switches or filters.
Given a simple enough shape, they often can be modelled as one-dimensional beams embedded in two or three dimensional space.

===Model description===

This model describes a slender beam which is actuated by a voltage between the beam and the ground electrode below (see <xr id="fig1"/>).

On the beam, at least three degrees of freedom per node have to be considered:

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|<math>y</math>
|Flexural displacement
|-
|<math>\theta_z</math>
|Flexural rotation
|-
|<math>q</math>
|Charge
|}

On the ground electrode, all spatial degrees of freedom are fixed, so only charge has to be considered.
The beam is supported either on the left side or on both sides.
The damping matrix is calculated by a linear combination of the mass matrix <math>M</math> and the stiffness matrix <math>K</math>.

The calculation of the electrostatic force would require a boundary element discretization, where it would be necessary to recalculate the capacity matrix for each time-step due to the motion of the charges
This would require an integration over the beam's elements.
This could be written in analytical form by using e.g. Gauss integration;
however, the complexity of the resulting system would be too high.
We therefore use the method shown in <ref name="siverberg1996"/>, i.e. we concentrate the charges on the nodes.
The capacity matrix then follows a simple <math>1/r</math> law.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>.

==Data==

Based on the finite element discretization presented in <ref name="weaver1990"/>, an interactive matrix generator has been created using [http://www.wolfram.com/ Wolfram Research]'s [http://www.wolfram.com/products/webmathematica webMathematica].
Models produced by this generator are in the DSIF<ref name="lienemsann2005"/> format, which allows for nonlinear terms.

All examples are made for a silicon beam with the following properties:

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
! Property
! Value
|-
|Beam length (l)
|<math>10^{-5} m</math>
|-
|Beam height (h)
|<math>10^{-6} m</math>
|-
|Beam width (w)
|<math>15 10^{-6} m</math>
|-
|Distance between beams (s)
|<math>200 10^{-9} m</math>
|-
|Material density (rho)
|<math>2330 kg/m^3</math>
|-
|Cross-sectional area (A)
|<math>150e-12 m12</math>
|-
|Moment of inertia (I)
|<math>1.25e-21 m^4</math>
|-
|Modulus of elasticity (E)
|<math>1.31e11 Pa</math>
|-
|Contribution of <math>M</math> to damping
|<math>0</math>
|-
|Contribution of <math>K</math> to damping
|<math>10^{-6}</math>
|-
|Support
|Both sides, <math>y</math> DOF only
|}

The following examples are available (all files are zipped compressed DSIF files, Units: SI):

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
! File
! Number of Nodes
! Number of Equations
! Compressed Size [kB]
|-
|[https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Eletrostatic%20beam%20%2838882%29/files/fileinnercontentproxy.2010-02-08.3711017358 E10.zip]
|10
|38
|4144
|-
|[https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Eletrostatic%20beam%20%2838882%29/files/fileinnercontentproxy.2010-02-08.3729192050 E100.zip]
|100
|398
|347679
|}

The <tt>.m</tt> files contain matrices <math>M</math>, <math>E</math>, <math>K</math>, <math>B</math>, <math>F</math> and <math>C</math>, the vector <math>f</math> and initial conditions for the following system of equations:

:<math>
\begin{align}
M \ddot{x}(t) + E\dot{x}(t) + K x(t) &= B u(t) + F f(x(t),u(t)) \\
y(t) &= C x(t)
\end{align}
</math>

where <math>B</math> is a <math>n \times 1</math> matrices with <math>1</math> at all charge DOFs of the upper beam and <math>C</math> is a <math>1 \times n</math> matrix with the only nonzero entry at the <math>y</math> DOF of the middle node.

Details of the implementation are available in a separate report ([https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Eletrostatic%20beam%20%2838882%29/files/fileinnercontentproxy.2010-02-08.3748527563 ElectrostatBeam.pdf]), see also <ref name="lienemann2006"/>.

A typical input to this system is a step response; periodic on/off switching is also possible.
The reduced model should thus both represent the step response as well as the possible influence of higher order harmonics.

==Dimensions==

System structure:

:<math>
\begin{align}
M \ddot{x}(t) + E\dot{x}(t) + K x(t) &= B u(t) + F f(x(t)) \\
y(t) &= C x(t)
\end{align}
</math>

System dimensions:

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

System variants:
<tt>E10</tt>: <math>N = 38</math>, <math>S = 28</math>,
<tt>E100</tt>: <math>N = 398</math>, <math>S = 298</math>.

==References==

<references>

<ref name="korvink2005"> J.G. Korvink, E.B. Rudnyi, [https://doi.org/10.1007/3-540-27909-1_11 Oberwolfach Benchmark Collection], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 311--315, 2005.</ref>

<ref name="lienemsann2005"> J. Lienemann, B. Salimbahrami, B. Lohmann, J.G. Korvink, [https://doi.org/10.1007/3-540-27909-1_12 A File Format for the Exchange of Nonlinear Dynamical ODE Systems], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 317--326, 2005.</ref>

<ref name="siverberg1996">L. Silverberg, L. Weaver, Jr., [https://doi.org/10.1115/1.2788876 Dynamics and Control of Electrostatic Structures], Journal of Applied Mechanics, Vol. 63, p. 383--391, 1996.</ref>

<ref name="weaver1990">W. Weaver, Jr., S.P. Timoshenko, D.H. Young, [https://books.google.de/books?id=YZ7t8LgRqi0C&lpg=PP1&dq=editions%3AgXRPzjrsO3IC&pg=PP1#v=onepage&q&f=false Vibration problems in engineering], 5th ed., Wiley, 1990.</ref>

<ref name="lienemann2006">J. Lienemann, E.B. Rudnyi, J.G. Korvink, [https://doi.org/10.1016/j.laa.2005.04.002 MST MEMS model order reduction: Requirements and benchmarks], Linear Algebra and its Applications, 415(2--3): 469--498 , 2006.</ref>

</references>