MOR Wiki - User contributions [en]

Talk:Butterfly Gyroscope

2018-04-25T11:42:59Z

Hund:

Talk:Butterfly Gyroscope

2018-04-24T13:08:50Z

Hund: /* Time domain simulation */ new section

Remark: Files gyro.[BCKM] in gyro.tar.gz are all in valid Matrix Market format, can be read by mmread for Matlab, but not by scipy.io.mmread (due to two blank lines, one below the comment section and one at the end). --[[User:Mlinaric|Mlinaric]] ([[User talk:Mlinaric|talk]]) 09:52, 4 April 2018 (CEST)

== Time domain simulation ==

I am not able to reproduce the oscillating trajectory as in the paper ''MEMS Compact Modeling Meets Model Order Reduction: Examples of the Application of Arnoldi Methods to Microsystem Devices'' by J. Lienemann et al.
Does anybody has experience with that?

Talk:Butterfly Gyroscope

2018-04-24T13:00:17Z

Hund:

Butterfly Gyroscope

2018-04-06T08:39:12Z

Hund: /* Citation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:ODE]]
[[Category:Linear]]
[[Category:Time invariant]]
[[Category:Second differential order]]
[[Category:MIMO]]
[[Category:Sparse]]

==Description==
<figure id="fig1">[[File:Butterfly1.jpg|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Butterfly2.jpg|490px|thumb|right|Figure 2]]</figure>
<figure id="fig3">[[File:Butterfly3.jpg|490px|thumb|right|Figure 3]]</figure>

The '''butterfly gyro''' is developed at the [http://www.imego.com Imego Institute] in an ongoing project with [[wikipedia:Saab_Bofors_Dynamics|Saab Bofors Dynamics AB]].
The butterfly is a vibrating micro-mechanical gyro that has sufficient theoretical performance characteristics to make it a promising candidate for use in inertial navigation applications.
The goal of the current project is to develop a micro unit for inertial navigation that can be commercialized in the high-end segment of the rate sensor market.
This project has reached the final stage of a three-year phase where the development and research efforts have ranged from model based signal processing, via electronics packaging to design and prototype manufacturing of the sensor element.
The project has also included the manufacturing of an [[wikipedia:Application-specific_integrated_circuit|ASIC]], named µSIC, that has been especially designed for the sensor (see <xr id="fig1"/>).

The gyro chip consists of a three-layer silicon wafer stack, in which the middle layer contains the sensor element.
The sensor consists of two wing pairs that are connected to a common frame by a set of beam elements (see <xr id="fig2"/> and <xr id="fig3"/>);
this is the reason the gyro is called the butterfly.
Since the structure is manufactured using an anisotropic wet-etch process, the connecting beams are slanted.
This makes it possible to keep all electrodes, both for capacitive excitation and detection, confined to one layer beneath the two wing pairs.
The excitation electrodes are the smaller dashed areas shown in <xr id="fig2"/>.
The detection electrodes correspond to the four larger ones.
By applying DC-biased AC-voltages to the four pairs of small electrodes, the wings are forced to vibrate in anti-phase in the wafer plane.
This is the excitation mode.
As the structure rotates about the axis of sensitivity (see <xr id="fig2"/>), each of the masses will be affected by a Coriolis acceleration.
This acceleration can be represented as an inertial force that is applied at right angles with the external angular velocity and the direction of motion of the mass. The Coriolis force induces an anti-phase motion of the wings out of the wafer plane.
This is the detection mode. The external angular velocity can be related to the amplitude of the detection mode, which is measured via the large electrodes.

When planning for and making decisions on future improvements of the butterfly, it is of importance to improve the efficiency of the gyro simulations.
Repeated analyses of the sensor structure have to be conducted with respect to a number of important issues.
Examples of such are sensitivity to shock, linear and angular vibration sensitivity, reaction to large rates and/or acceleration, different types of excitation load cases and the effect of force-feedback.

The use of model order reduction indeed decreases runtimes for repeated simulations.
Moreover, the reduction technique enables a transformation of the FE representation of the gyro into a state space equivalent formulation.
This will prove helpful in testing the model based Kalman signal processing algorithms that are being designed for the '''butterfly gyro'''.

The structural model of the gyroscope has been done in [http://www.ansys.com/ ANSYS] using quadratic tetrahedral elements (SOLID187, see <xr id="fig3"/>).
The model shown is a simplified one with a coarse mesh as it is designed to test the model reduction approaches.
It includes the pure structural mechanics problem only. The load vector is composed from time-varying nodal forces applied at the centers of the excitation electrodes (see <xr id="fig2"/>).
The amplitude and frequency of each force is equal to <math>0.055 \mu N</math> and <math>2384 Hz</math>, respectively.
The Dirichlet boundary conditions have been applied to all degree of freedom of the nodes belonging to the top and bottom surfaces of the frame.
The output nodes are listed in Table 2 and correspond to the centers of the detection electrodes (see <xr id="fig3"/>).
The structural model

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

contains the mass <math>M</math> and stiffness matrices <math>K</math>.
The damping matrix <math>E</math> can be modeled as <math>E = \alpha M + \beta K</math>, where the typical values of <math>\alpha</math> and <math>\beta</math> are <math>0</math> and <math>10^{-6}</math> respectively.
The nature of the damping matrix is in reality more complex (squeeze film damping, thermo-elastic damping, etc.) but this simple approach has been chosen with respect to the model reduction test.
<math>B</math> is the load vector, <math>C</math> is the output matrix.

The statistics for the matrices is shown in Table 1.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: System matrices for the gyroscope.''
|-
!Matrix
!m
!n
!nnz
!Is Symmetric?
|-
|M
|17361
|17361
|178896
|Yes
|-
|K
|17361
|17361
|519260
|Yes
|-
|B
|17361
|1
|8
|No
|-
|C
|12
|17361
|12
|No
|}

The outputs are detailed in Table 2.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 2. Outputs for the Butterfly Gyro Model.''
|-
!Index
!Code
!Comment
|-
|1-3
|det1m_Ux, det1m_Uy, det1m_Uz
|Displacements of detection electrode 1, (bottom left large electrode of Fig. 2)
|-
|4-6
|det1p_Ux, det1p_Uy, det1p_Uz
|Displacements of detection electrode 2, (bottom right large electrode of Fig. 2)
|-
|7-9
|det2m_Ux, det2m_Uy, det2m_Uz
|Displacements of detection electrode 3, (top left large electrode of Fig. 2)
|-
|10-12
|det2p_Ux, det2p_Uy, det2p_Uz
|Displacements of detection electrode 4, (top right large electrode of Fig. 2)
|}

The model reduction of the gyroscope model by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem] is described in
<ref name="lienemann04"/>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 35889, see: <ref name="billger05"/>.

==Data==

Download matrices in [http://math.nist.gov/MatrixMarket/ Matrix Market] format:
* [https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/The%20Butterfly%20Gyro%20%2835889%29/files/fileinnercontentproxy.2010-01-31.7493503105 gyro.tar.gz] (7.4 MB)

The matrix name is used as an extension of the matrix file. File <tt>*.C.names</tt> contains a list of output names written consecutively.

==Dimensions==

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

System dimensions:

<math>M \in \mathbb{R}^{17361 \times 17361}</math>,
<math>E \in \mathbb{R}^{17361 \times 17361}</math>,
<math>K \in \mathbb{R}^{17361 \times 17361}</math>,
<math>B \in \mathbb{R}^{17361}</math>,
<math>C \in \mathbb{R}^{12 \times 17361}</math>

==Citation==
To cite this benchmark, use the following references:

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection '''Butterfly Gyroscope'''. hosted at MORwiki - Model Order Reduction Wiki, 2004. http://modelreduction.org/index.php/Butterfly_Gyroscope

@MISC{morwiki_gyro,
author = {Oberwolfach Benchmark Collection},
title = {Butterfly Gyroscope},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = {<nowiki>http://modelreduction.org/index.php/Butterfly_Gyroscope</nowiki>},
year = 2004
}

* For the background on the benchmark:

@INPROCEEDINGS{morBil05,
author = {D. Billger},
title = {The Butterfly Gyro},
booktitle = {Dimension Reduction of Large-Scale Systems},
publisher = {Springer-Verlag, Berlin/Heidelberg, Germany},
year = 2005,
volume = 45,
pages = {349--352},
series = Lecture Notes in Computational Science and Engineering,
doi = {10.1007/3-540-27909-1_18}
}

==References==

<references>

<ref name="lienemann04"> J. Lienemann, D. Billger, E.B. Rudnyi, A. Greiner, and J.G. Korvink, [http://modelreduction.com/doc/papers/lienemann04MSM.pdf MEMS Compact Modeling Meets Model Order Reduction: Examples of the Application of Arnoldi Methods to Microsystem Devices], Technical Proceedings of the 2004 Nanotechnology Conference and Trade Show, Nanotech 2004, March 7-11, 2004, Boston, Massachusetts, USA.</ref>

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

<ref name="billger05"> D. Billger, [https://doi.org/10.1007/3-540-27909-1_18 The Butterfly Gyro], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 349--352, 2005.</ref>

</references>

Butterfly Gyroscope

2018-04-06T08:22:14Z

Hund:

{{preliminary}} 

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:ODE]]
[[Category:Linear]]
[[Category:Time invariant]]
[[Category:Second differential order]]
[[Category:MIMO]]
[[Category:Sparse]]

==Description==
<figure id="fig1">[[File:Butterfly1.jpg|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Butterfly2.jpg|490px|thumb|right|Figure 2]]</figure>
<figure id="fig3">[[File:Butterfly3.jpg|490px|thumb|right|Figure 3]]</figure>

The '''butterfly gyro''' is developed at the [http://www.imego.com Imego Institute] in an ongoing project with [[wikipedia:Saab_Bofors_Dynamics|Saab Bofors Dynamics AB]].
The butterfly is a vibrating micro-mechanical gyro that has sufficient theoretical performance characteristics to make it a promising candidate for use in inertial navigation applications.
The goal of the current project is to develop a micro unit for inertial navigation that can be commercialized in the high-end segment of the rate sensor market.
This project has reached the final stage of a three-year phase where the development and research efforts have ranged from model based signal processing, via electronics packaging to design and prototype manufacturing of the sensor element.
The project has also included the manufacturing of an [[wikipedia:Application-specific_integrated_circuit|ASIC]], named µSIC, that has been especially designed for the sensor (see <xr id="fig1"/>).

The gyro chip consists of a three-layer silicon wafer stack, in which the middle layer contains the sensor element.
The sensor consists of two wing pairs that are connected to a common frame by a set of beam elements (see <xr id="fig2"/> and <xr id="fig3"/>);
this is the reason the gyro is called the butterfly.
Since the structure is manufactured using an anisotropic wet-etch process, the connecting beams are slanted.
This makes it possible to keep all electrodes, both for capacitive excitation and detection, confined to one layer beneath the two wing pairs.
The excitation electrodes are the smaller dashed areas shown in <xr id="fig2"/>.
The detection electrodes correspond to the four larger ones.
By applying DC-biased AC-voltages to the four pairs of small electrodes, the wings are forced to vibrate in anti-phase in the wafer plane.
This is the excitation mode.
As the structure rotates about the axis of sensitivity (see <xr id="fig2"/>), each of the masses will be affected by a Coriolis acceleration.
This acceleration can be represented as an inertial force that is applied at right angles with the external angular velocity and the direction of motion of the mass. The Coriolis force induces an anti-phase motion of the wings out of the wafer plane.
This is the detection mode. The external angular velocity can be related to the amplitude of the detection mode, which is measured via the large electrodes.

When planning for and making decisions on future improvements of the butterfly, it is of importance to improve the efficiency of the gyro simulations.
Repeated analyses of the sensor structure have to be conducted with respect to a number of important issues.
Examples of such are sensitivity to shock, linear and angular vibration sensitivity, reaction to large rates and/or acceleration, different types of excitation load cases and the effect of force-feedback.

The use of model order reduction indeed decreases runtimes for repeated simulations.
Moreover, the reduction technique enables a transformation of the FE representation of the gyro into a state space equivalent formulation.
This will prove helpful in testing the model based Kalman signal processing algorithms that are being designed for the '''butterfly gyro'''.

The structural model of the gyroscope has been done in [http://www.ansys.com/ ANSYS] using quadratic tetrahedral elements (SOLID187, see <xr id="fig3"/>).
The model shown is a simplified one with a coarse mesh as it is designed to test the model reduction approaches.
It includes the pure structural mechanics problem only. The load vector is composed from time-varying nodal forces applied at the centers of the excitation electrodes (see <xr id="fig2"/>).
The amplitude and frequency of each force is equal to <math>0.055 \mu N</math> and <math>2384 Hz</math>, respectively.
The Dirichlet boundary conditions have been applied to all degree of freedom of the nodes belonging to the top and bottom surfaces of the frame.
The output nodes are listed in Table 2 and correspond to the centers of the detection electrodes (see <xr id="fig3"/>).
The structural model

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

contains the mass <math>M</math> and stiffness matrices <math>K</math>.
The damping matrix <math>E</math> can be modeled as <math>E = \alpha M + \beta K</math>, where the typical values of <math>\alpha</math> and <math>\beta</math> are <math>0</math> and <math>10^{-6}</math> respectively.
The nature of the damping matrix is in reality more complex (squeeze film damping, thermo-elastic damping, etc.) but this simple approach has been chosen with respect to the model reduction test.
<math>B</math> is the load vector, <math>C</math> is the output matrix.

The statistics for the matrices is shown in Table 1.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: System matrices for the gyroscope.''
|-
!Matrix
!m
!n
!nnz
!Is Symmetric?
|-
|M
|17361
|17361
|178896
|Yes
|-
|K
|17361
|17361
|519260
|Yes
|-
|B
|17361
|1
|8
|No
|-
|C
|12
|17361
|12
|No
|}

The outputs are detailed in Table 2.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 2. Outputs for the Butterfly Gyro Model.''
|-
!Index
!Code
!Comment
|-
|1-3
|det1m_Ux, det1m_Uy, det1m_Uz
|Displacements of detection electrode 1, (bottom left large electrode of Fig. 2)
|-
|4-6
|det1p_Ux, det1p_Uy, det1p_Uz
|Displacements of detection electrode 2, (bottom right large electrode of Fig. 2)
|-
|7-9
|det2m_Ux, det2m_Uy, det2m_Uz
|Displacements of detection electrode 3, (top left large electrode of Fig. 2)
|-
|10-12
|det2p_Ux, det2p_Uy, det2p_Uz
|Displacements of detection electrode 4, (top right large electrode of Fig. 2)
|}

The model reduction of the gyroscope model by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem] is described in
<ref name="lienemann04"/>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 35889, see: <ref name="billger05"/>.

==Data==

Download matrices in [http://math.nist.gov/MatrixMarket/ Matrix Market] format:
* [https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/The%20Butterfly%20Gyro%20%2835889%29/files/fileinnercontentproxy.2010-01-31.7493503105 gyro.tar.gz] (7.4 MB)

The matrix name is used as an extension of the matrix file. File <tt>*.C.names</tt> contains a list of output names written consecutively.

==Dimensions==

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

System dimensions:

<math>M \in \mathbb{R}^{17361 \times 17361}</math>,
<math>E \in \mathbb{R}^{17361 \times 17361}</math>,
<math>K \in \mathbb{R}^{17361 \times 17361}</math>,
<math>B \in \mathbb{R}^{17361}</math>,
<math>C \in \mathbb{R}^{12 \times 17361}</math>

==Citation==
To cite this benchmark, use the following references:

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection '''Butterfly Gyroscope'''. hosted at MORwiki - Model Order Reduction Wiki, 2004. http://modelreduction.org/index.php/Butterfly_Gyroscope

@MISC{morwiki_gyro,
author = {Oberwolfach Benchmark Collection},
title = {Butterfly Gyroscope},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = {<nowiki>http://modelreduction.org/index.php/Butterfly_Gyroscope</nowiki>},
year = 2004
}

* For the background on the benchmark:

@INPROCEEDINGS{morLieBRetal04,
title = {MEMS Compact Modeling Meets Model Order Reduction: Examples of the Application of Arnoldi Methods to Microsystem Devices},
author = {J. Lienemann, D. Billger, E.B. Rudnyi, A. Greiner, and J.G. Korvink},
booktitle = {Technical Proceedings of the 2004 Nanotechnology Conference and Trade Show}
year = 2004,
month = {March},
pages = {303--306},
month = {June},
publisher = {Nano Science and Technology Institute, Boston},
isbn = {0-9728422-8-4}
}
==References==

<references>

<ref name="lienemann04"> J. Lienemann, D. Billger, E.B. Rudnyi, A. Greiner, and J.G. Korvink, [http://modelreduction.com/doc/papers/lienemann04MSM.pdf MEMS Compact Modeling Meets Model Order Reduction: Examples of the Application of Arnoldi Methods to Microsystem Devices], Technical Proceedings of the 2004 Nanotechnology Conference and Trade Show, Nanotech 2004, March 7-11, 2004, Boston, Massachusetts, USA.</ref>

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

<ref name="billger05"> D. Billger, [https://doi.org/10.1007/3-540-27909-1_18 The Butterfly Gyro], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 349--352, 2005.</ref>

</references>

Linear 1D Beam

2018-04-06T08:09:55Z

Hund:

{{preliminary}} 

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:ODE]]
[[Category:Linear]]
[[Category:Time invariant]]
[[Category:Second differential order]]
[[Category:SISO]]
[[Category:Sparse]]

==Description==
<figure id="fig1">[[File:Beam1.gif|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Beam2.gif|490px|thumb|right|Figure 2]]</figure>
<figure id="fig3">[[File:Beam3.gif|490px|thumb|right|Figure 3]]</figure>
<figure id="fig4">[[File:Beam4.gif|490px|thumb|right|Figure 4]]</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 with four degrees of freedom per node:

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

See <xr id="fig2"/> for Degree of Freedom <math>x</math>, <xr id="fig3"/> for Degree of Freedom <math>\theta_x</math> and <xr id="fig4"/> for Degrees of freedom <math>y</math> and <math>\theta_z</math>.

The beam is supported either on the left side or on both sides. For the left side (fixed) support,
the force is applied on the rightmost node in <math>y</math> direction, whereas for the support on both sides (simply supported), a node in the middle is loaded.
The damping matrix is calculated by a linear combination of the mass matrix <math>M</math> and the stiffness matrix <math>K</math>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38861.

==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].
However, models produced by this generator are in the <tt>DSIF</tt><ref name="lienemsann2005"/> format, which allows for nonlinear terms.
For the purpose of the benchmark collection, we have precomputed four systems and converted them to the [http://math.nist.gov/MatrixMarket/ Matrix market] format which is easier to import in standard computer algebra packages.

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

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Property
|Value
|-
|Beam length (l)
|<math>0.1</math> m
|-
|Material density (rho)
|<math>8000</math> kg/m3
|-
|Cross-sectional area (A)
|<math>7.854\cdot 10^{-7}</math> m2
|-
|Moment of inertia (I)
|<math>4.909\cdot 10^{-14}</math> m4
|-
|Polar moment of inertia (J)
|<math>9.817\cdot 10^{-14}</math>
|-
|Modulus of elasticity (E)
|<math>2\cdot 10^{11}</math> Pa
|-
|Poisson ratio (nu)
|<math>0.29</math>
|-
|Contribution of <math>M</math> to damping
|<math>100</math>
|-
|Contribution of <math>K</math> to damping
|<math>0.01</math>
|-
|Support
|Simple, both sides
|}

The following examples are available (all files are compressed <tt>.zip</tt> archives, Units: SI):

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|File
|Degrees of freedom
|Number of nodes
|Number of equations
|File size [B]
|Compressed size [B]
|-
|[https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Linear%201D%20Beam%20%2838861%29/files/fileinnercontentproxy.2010-01-31.3349728177 LF10.zip]
|flexural (<math>y</math> and <math>\theta_z</math>)
|10
|18
|5935
|2384
|-
|[https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Linear%201D%20Beam%20%2838861%29/files/fileinnercontentproxy.2010-01-31.3358570716 LF10000.zip]
|flexural (<math>y</math> and <math>\theta_z</math>)
|10000
|19998
|6640324
|716807
|-
|[https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Linear%201D%20Beam%20%2838861%29/files/fileinnercontentproxy.2010-01-31.3367011092 LFAT5.zip]
|flexural (<math>y</math> and <math>\theta_z</math>), axial, torsional
|5
|14
|4045
|2255
|-
|[https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Linear%201D%20Beam%20%2838861%29/files/fileinnercontentproxy.2010-01-31.3373723032 LFAT5000.zip]
|flexural (<math>y</math> and <math>\theta_z</math>), axial, torsional
|50000
|19994
|5532532
|627991
|}

The zip files contain matrices <math>M</math>, <math>E</math>, <math>K</math>, <math>B</math> and <math>C</math> for the following system of equations:

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

where <math>B</math> is a <math>n \times 1</math> matrix 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 [https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Linear%201D%20Beam%20%2838861%29/files/fileinnercontentproxy.2010-01-31.3407950776 report]. 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.
See also <ref name="lienemann2006"/>.

==Dimensions==

System structure:

:<math>
\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(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>C \in \mathbb{R}^{1 \times N}</math>

System variants:

<tt>LF10</tt>: <math>N = 18</math>,
<tt>LF100</tt>: <math>N = 19998</math>,
<tt>LFAT5</tt>: <math>N = 14</math>,
<tt>LFAT5000</tt>: <math>N = 19994</math>,

==Citation==
To cite this benchmark, use the following references:

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection '''Linear 1D Beam'''. hosted at MORwiki - Model Order Reduction Wiki, 2004. http://modelreduction.org/index.php/Linear_1D_Beam

@MISC{morwiki_linear_beam,
author = {Oberwolfach Benchmark Collection},
title = {Linear 1{D} Beam},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = {<nowiki>http://modelreduction.org/index.php/Linear_1D_Beam</nowiki>},
year = 2004
}

* For the background on the benchmark:

@TechReport{morLieRK06,
title = {MST MEMS Model Order Reduction: Requirements and Benchmarks},
author = {J. Lienemann, E.B. Rudnyi and J.G. Korvink},
journal = {Linear Algebra and its Applications},
year = 2006,
volume = 415,
issue = 2--3,
pages = {469--498},
month = {June},
publisher = {Elsevier},
doi = {10.1016/j.laa.2005.04.002}
}

==References==

<references>

<ref name="korvink2005"> J.G. Korvink, E.B. Rudnyi, [https://doi.org/10.1007/3-540-27909-1_11 Oberwolfach Benchmark Collection], In: 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="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>

Linear 1D Beam

2018-04-06T08:09:21Z

Hund:

{{preliminary}} 

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:ODE]]
[[Category:Linear]]
[[Category:Time invariant]]
[[Category:Second differential order]]
[[Category:SISO]]
[[Category:Sparse]]

==Description==
<figure id="fig1">[[File:Beam1.gif|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Beam2.gif|490px|thumb|right|Figure 2]]</figure>
<figure id="fig3">[[File:Beam3.gif|490px|thumb|right|Figure 3]]</figure>
<figure id="fig4">[[File:Beam4.gif|490px|thumb|right|Figure 4]]</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 with four degrees of freedom per node:

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

See <xr id="fig2"/> for Degree of Freedom <math>x</math>, <xr id="fig3"/> for Degree of Freedom <math>\theta_x</math> and <xr id="fig4"/> for Degrees of freedom <math>y</math> and <math>\theta_z</math>.

The beam is supported either on the left side or on both sides. For the left side (fixed) support,
the force is applied on the rightmost node in <math>y</math> direction, whereas for the support on both sides (simply supported), a node in the middle is loaded.
The damping matrix is calculated by a linear combination of the mass matrix <math>M</math> and the stiffness matrix <math>K</math>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38861.

==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].
However, models produced by this generator are in the <tt>DSIF</tt><ref name="lienemsann2005"/> format, which allows for nonlinear terms.
For the purpose of the benchmark collection, we have precomputed four systems and converted them to the [http://math.nist.gov/MatrixMarket/ Matrix market] format which is easier to import in standard computer algebra packages.

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

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Property
|Value
|-
|Beam length (l)
|<math>0.1</math> m
|-
|Material density (rho)
|<math>8000</math> kg/m3
|-
|Cross-sectional area (A)
|<math>7.854\cdot 10^{-7}</math> m2
|-
|Moment of inertia (I)
|<math>4.909\cdot 10^{-14}</math> m4
|-
|Polar moment of inertia (J)
|<math>9.817\cdot 10^{-14}</math>
|-
|Modulus of elasticity (E)
|<math>2\cdot 10^{11}</math> Pa
|-
|Poisson ratio (nu)
|<math>0.29</math>
|-
|Contribution of <math>M</math> to damping
|<math>100</math>
|-
|Contribution of <math>K</math> to damping
|<math>0.01</math>
|-
|Support
|Simple, both sides
|}

The following examples are available (all files are compressed <tt>.zip</tt> archives, Units: SI):

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|File
|Degrees of freedom
|Number of nodes
|Number of equations
|File size [B]
|Compressed size [B]
|-
|[https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Linear%201D%20Beam%20%2838861%29/files/fileinnercontentproxy.2010-01-31.3349728177 LF10.zip]
|flexural (<math>y</math> and <math>\theta_z</math>)
|10
|18
|5935
|2384
|-
|[https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Linear%201D%20Beam%20%2838861%29/files/fileinnercontentproxy.2010-01-31.3358570716 LF10000.zip]
|flexural (<math>y</math> and <math>\theta_z</math>)
|10000
|19998
|6640324
|716807
|-
|[https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Linear%201D%20Beam%20%2838861%29/files/fileinnercontentproxy.2010-01-31.3367011092 LFAT5.zip]
|flexural (<math>y</math> and <math>\theta_z</math>), axial, torsional
|5
|14
|4045
|2255
|-
|[https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Linear%201D%20Beam%20%2838861%29/files/fileinnercontentproxy.2010-01-31.3373723032 LFAT5000.zip]
|flexural (<math>y</math> and <math>\theta_z</math>), axial, torsional
|50000
|19994
|5532532
|627991
|}

The zip files contain matrices <math>M</math>, <math>E</math>, <math>K</math>, <math>B</math> and <math>C</math> for the following system of equations:

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

where <math>B</math> is a <math>n \times 1</math> matrix 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 [https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Linear%201D%20Beam%20%2838861%29/files/fileinnercontentproxy.2010-01-31.3407950776 report]. 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.
See also <ref name="lienemann2006"/>.

==Dimensions==

System structure:

:<math>
\begin{align}
M \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(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>C \in \mathbb{R}^{1 \times N}</math>

System variants:

<tt>LF10</tt>: <math>N = 18</math>,
<tt>LF100</tt>: <math>N = 19998</math>,
<tt>LFAT5</tt>: <math>N = 14</math>,
<tt>LFAT5000</tt>: <math>N = 19994</math>,

==Citation==
To cite this benchmark, use the following references:

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection '''Linear 1D Beam'''. hosted at MORwiki - Model Order Reduction Wiki, 2004. http://modelreduction.org/index.php/Linear_1D_Beam

@MISC{morwiki_linear_beam,
author = {Oberwolfach Benchmark Collection},
title = {Linear 1{D} Beam},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = {<nowiki>http://modelreduction.org/index.php/Linear_1D_Beam</nowiki>},
year = 2004
}

* For the background on the benchmark:

@TechReport{morLieRK06,
title = {MST MEMS Model Order Reduction: Requirements and Benchmarks},
author = {J. Lienemann, E.B. Rudnyi and J.G. Korvink},
journal = {Linear Algebra and its Applications},
year = 2006,
volume = 415,
issue = 2--3,
pages = {469--498},
month = {June},
publisher = {Elsevier},
doi = {10.1016/j.laa.2005.04.002}
}

==References==

<references>

<ref name="korvink2005"> J.G. Korvink, E.B. Rudnyi, [https://doi.org/10.1007/3-540-27909-1_11 Oberwolfach Benchmark Collection], In: 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="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>