MOR Wiki - User contributions [en]

Glossary

2024-04-02T15:25:30Z

Lund: Update to match current MORB database

{{preliminary}} 

This page contains a glossary of terms related to the [[MORB]] database, the [[Search Tool]], and this wiki as a whole.

Note that all boolean fields are really quadruple valued; in addition to 0 (false) and 1 (true), other options include the following:
* an empty field, indicating a lack of knowledge, e.g., either no attempt has been made to figure out the value, or none of the available methods have been successful in determining the value thus far.
* NaN, indicating either that the property does not apply or is otherwise impossible to determine

{| class="wikitable" style="background-color:#FFF;"
|- style="font-weight:bold; vertical-align:middle;"
! Abbreviation / Attribute / Term
! Description
|- style="vertical-align:middle;"
| <code>comments</code>
| (char) internal comments
|- style="vertical-align:middle;"
| <code>dataSource</code>
| (char) where the data was pulled from during processing: [https://gitlab.mpi-magdeburg.mpg.de/himpe/unimat unimat], [https://sites.google.com/site/rommes/software rommes], or [https://modelreduction.org morwiki]
|- style="vertical-align:middle;"
| <code>sourceFilename</code>
| (char) filename of data used during processing
|- style="vertical-align:middle;"
| AP
| affine parametric; parameters of any component are linear combinations of fixed matrices
|- style="vertical-align:middle;"
| <code>components</code>
| (char) matrices stored in .mat file; see also [[Models]]
|- style="vertical-align:middle;"
| <code>cond*</code>
| (list of float) the numerically computed ratio between the largest and smallest singular values of the matrix * (including 0); Inf is treated as a float in MATLAB; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>benchmarkCreator</code>
| (list of char) ORCID(s) of original author(s) or creator(s) of dataset
|- style="vertical-align:middle;"
| DAE
| [https://en.wikipedia.org/wiki/Differential_algebraic_equation differential algebraic equation]
|- style="vertical-align:middle;"
| <code>daeDiffIndex</code>
| (int) DAE differentiation index; roughly the number of differentiations required to make a system an ODE
|- style="vertical-align:middle;"
| DEIM
| discrete empirical interpolation method
|- style="vertical-align:middle;"
| <code>category</code>
| (char) folder / collection to which a benchmark belongs
|- style="vertical-align:middle;"
| DoF
| [https://en.wikipedia.org/wiki/Degrees_of_Freedom degrees of freedom]
|- style="vertical-align:middle;"
| DSPMR
| [[Dominant_Subspaces|dominant subspace projection model reduction]]
|- style="vertical-align:middle;"
| <code>dataEditor</code>
| (list of char) ORCID(s) of editor(s) who converted data to new .mat format
|- style="vertical-align:middle;"
| FEM
| [https://en.wikipedia.org/wiki/Finite_Element_Method finite element method]
|- style="vertical-align:middle;"
| <code>filename</code>
| (char) filename and benchmark ID
|- style="vertical-align:middle;"
| FOM
| full order model
|- style="vertical-align:middle;"
| FOS
| first-order system
|- style="vertical-align:middle;"
| <code>is*CholAble</code>
| (list of bool) the matrix * is symmetric positive definite, i.e., it admits a Cholesky factorization (and is therefore "Cholesky-able"); note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>is*Sparse</code>
| (list of bool) the matrix * is stored in a sparse format; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>is*Symm</code>
| (list of bool) the matrix is symmetric; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>isContractive</code>
| (bool) a system that is dissipative w.r.t. <math>s(u(t),y(t)) = ||u(t)||^2 - ||y(t)||^2</math>
|- style="vertical-align:middle;"
| <code>isDAE</code>
| (bool) differential algebraic equation; an LTI DAE has a nonzero, singular E component
|- style="vertical-align:middle;"
| <code>isPassive</code>
| (bool) a system that is dissipative w.r.t s(u(t),y(t)) = u(t)^T y(t)
|- style="vertical-align:middle;"
| <code>isSquare</code>
| (bool) nInputs = nOutputs
|- style="vertical-align:middle;"
| <code>isStable</code>
| (int) numerically computed; 0 = no stability 1 = asymptotically stable; i.e., all finite eigenvalues of the pencil <math>sE-A</math> (<math>E = I</math>, if not otherwise specified) have negative real part; consequently, the solution of the corresponding LTI with <math>u(t) = 0</math> tends to 0 as <math>t \to \infty</math> 2 = ? 3 = ? (other stability definitions added and defined as necessary)
|- style="vertical-align:middle;"
| <code>isStateSpaceSymm</code>
| (bool) state-space symmetric: <math>A = A^T</math>, <math>C = B^T</math>, <math>E = E^T</math>, <math>D = D^T</math>; trivially implies <code>isSysSymm</code>; similarly defined for SOS, but require instead that <math>M = M^T</math>, <math>C_v = 0, C_p = B^T</math>, and <math>K = K^T</math>
|- style="vertical-align:middle;"
| <code>isSysSymm</code>
| (bool) <math>H(s) = C (sE -A)^{-1} B+D</math> is symmetric; equivalently, the system's Markov parameters are symmetric For a distinction between symmetry and state-space symmetry, see Def 2.1 in [https://www.sciencedirect.com/science/article/pii/S0167691198000243 this paper]. [https://doi.org/10.1109/PROC.1983.12688 Another useful source]
|- style="vertical-align:middle;"
| <code>license</code>
| (char) license associated to data
|- style="vertical-align:middle;"
| LTI
| linear time-invariant
|- style="vertical-align:middle;"
| LTV
| linear time-varying
|- style="vertical-align:middle;"
| MIMO
| [https://en.wikipedia.org/wiki/MIMO multiple-input, multiple-output]
|- style="vertical-align:middle;"
| MOR
| [https://modelreduction.org/ model order reduction]
|- style="vertical-align:middle;"
| <code>MORWikiLink</code>
| (char) link to MORWiki page
|- style="vertical-align:middle;"
| <code>MORWikiPageName</code>
| (char) fixed page name with plain text description of system in the MORWiki
|- style="vertical-align:middle;"
| <code>nInputs</code>
| (int) number of inputs
|- style="vertical-align:middle;"
| NL
| nonlinear; a nonlinear system depends nonlinearly on <math>x(t)</math>
|- style="vertical-align:middle;"
| <code>nnz*</code>
| (list of int) when the matrix * is sparse, the number of nonzeros; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>nOutputs</code>
| (int) number of outputs
|- style="vertical-align:middle;"
| <code>nParameters</code>
| (int) maximum number of parameters in a parametric system
|- style="vertical-align:middle;"
| <code>nStates</code>
| (int) number of states
|- style="vertical-align:middle;"
| <code>nUnstabPoles</code>
| (int) the number of poles with nonnegative real part, computed numerically
|- style="vertical-align:middle;"
| ODE
| [https://en.wikipedia.org/wiki/Ordinary_differential_equation ordinary differential equation]
|- style="vertical-align:middle;"
| PDE
| [https://en.wikipedia.org/wiki/Partial_differential_equation partial differential equation]
|- style="vertical-align:middle;"
| PMOR
| parametric model order reduction
|- style="vertical-align:middle;"
| POD
| proper orthogonal decomposition
|- style="vertical-align:middle;"
| QBS
| quadratic bilinear system
|- style="vertical-align:middle;"
| RBM
| [https://pnp.mathematik.uni-stuttgart.de/ians/haasdonk/publications/RBtutorial_preprint_update_with_header.pdf reduced basis method]
|- style="vertical-align:middle;"
| ROM
| reduced order model
|- style="vertical-align:middle;"
| SISO
| [https://en.wikipedia.org/wiki/Single-Input_and_Single-Output single-input, single-output]
|- style="vertical-align:middle;"
| SOS
| second-order system
|- style="vertical-align:middle;"
| SVD
| [https://en.wikipedia.org/wiki/Singular_Value_Decomposition singular value decomposition]
|- style="vertical-align:middle;"
| <code>systemClass</code>
| (char) model realization; see also [[Models]]
|- style="vertical-align:middle;"
| <code>zenodoDOI</code>
| (char) zenodo DOI (not the full URL!)
|}

Transmission Lines (SLICOT)

2023-11-23T13:26:07Z

Lund: /* Citation */ Correct DOI typo

[[Category:benchmark]]
[[Category:SLICOT]]
[[Category:MIMO]]
[[Category:Dense]]
[[Category:Differential_algebraic_system]]

==Description: Model of a Transmission Line Model==

This benchmark models impedence of interconnected structures of a [[wikipedia:Transmission_line|transmission line]].
More details can be found in <ref name="li99"/>, <ref name="marques98"/> and <ref name="chahlaoui02"/>.

For a larger transmission lines model see the alternative [[Transmission_Lines]] benchmark.

==Origin==

This benchmark is part of the '''SLICOT Benchmark Examples for Model Reduction'''<ref name="chahlaoui05"/>.

==Data==

The system matrices <math>A</math>, <math>B</math>, <math>C</math>, <math>E</math> are available from the [http://slicot.org/20-site/126-benchmark-examples-for-model-reduction SLICOT benchmarks] page: [http://slicot.org/objects/software/shared/bench-data/tline.zip tline.zip] and are stored as MATLAB [https://www.mathworks.com/help/matlab/import_export/mat-file-versions.html .mat] file.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
E\dot{x}(t) &=& Ax(t) + Bu(t) \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>A \in \mathbb{R}^{256 \times 256}</math>,
<math>B \in \mathbb{R}^{256 \times 2}</math>,
<math>C \in \mathbb{R}^{2 \times 256}</math>,
<math>E \in \mathbb{R}^{256 \times 256}</math>

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::Niconet e.V., '''SLICOT - Subroutine Library in Systems and Control Theory''', http://www.slicot.org

@MANUAL{slicot_tline,
title = {{SLICOT} - Subroutine Library in Systems and Control Theory},
organization = {Niconet e.V.}
address = <nowiki>{\url{http://www.slicot.org}}</nowiki>,
key = {SLICOT}
}

* For the background on the benchmark:

@INPROCEEDINGS{MarKWetal98,
author = <nowiki>{N. Marques and M. Kamon and J. White and L.M. Silveira}</nowiki>,
title = {A mixed nodal-mesh formulation for efficient extraction and passive reduced-order modeling of 3D interconnects},
booktitle = {Proceedings of the 1998 Design and Automation Conference},
pages = {297--302},
year = {1998},
doi = {10.1145/277044.277132}
}

==References==

<references>

<ref name="li99"> J.R. Li, J. White. [https://doi.org/10.1109/ICCAD.1999.810679 Efficient model reduction of interconnect via approximate system gramians]. 1999 IEEE/ACM International Conference on Computer-Aided Design: 380--383, 1999.</ref>

<ref name="marques98"> N. Marques, M. Kamon, J. White, L.M. Silveira. [https://doi.org/10.1145/277044.277132 A mixed nodal-mesh formulation for efficient extraction and passive reduced-order modeling of 3D interconnects]. Proceedings of the 1998 Design and Automation Conference: 297--302, 1998.</ref>

<ref name="chahlaoui02"> Y. Chahlaoui, P. Van Dooren, [http://eprints.maths.manchester.ac.uk/1040/1/ChahlaouiV02a.pdf A collection of Benchmark examples for model reduction of linear time invariant dynamical systems], Working Note 2002-2: 2002.</ref>

<ref name="chahlaoui05"> Y. Chahlaoui, P. Van Dooren, [https://doi.org/10.1007/3-540-27909-1_24 Benchmark Examples for Model Reduction of Linear Time-Invariant Dynamical Systems], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 379--392, 2005.</ref>

</references>

Talk:Thermal Model

2023-11-21T08:50:40Z

Lund: Created page with "The link to the background paper "Parametric Model Reduction to Generate Boundary Condition Independent Compact Thermal Model" is broken. --KLund"

The link to the background paper "Parametric Model Reduction to Generate Boundary Condition Independent Compact Thermal Model" is broken. --KLund

Talk:Nonlinear Heat Transfer

2023-11-20T16:52:31Z

Lund: Created page with "Should background resource be updated with https://link.springer.com/chapter/10.1007/3-540-27909-1_13 ? --KLund"

Should background resource be updated with https://link.springer.com/chapter/10.1007/3-540-27909-1_13 ? --KLund

Talk:Convective Thermal Flow

2023-11-20T16:14:36Z

Lund:

- Permalink to background paper is lost. However, I could find it here: https://www.semanticscholar.org/paper/Model-Order-Reduction-for-Linear-Convective-Thermal-Moosmann-Rudnyi/58a0b553d6a7c59c2bcf190bc0bddda80496ff4c - KLund

- The description needs to include the PDE description from [5]. - C.H.

Talk:Convective Thermal Flow

2023-11-20T16:14:09Z

Lund:

Permalink to background paper is lost. However, I could find it here: https://www.semanticscholar.org/paper/Model-Order-Reduction-for-Linear-Convective-Thermal-Moosmann-Rudnyi/58a0b553d6a7c59c2bcf190bc0bddda80496ff4c - KLund
The description needs to include the PDE description from [5]. - C.H.

MORB

2023-09-14T14:48:54Z

Lund: add citations and updated links for paper and code

A model order reduction benchmark (MORB) tool is currently under development. It is a demonstrator of the generic benchmark framework (MaRDIMark) developed for Task Area 2 of the Mathematical Research Data Initiative ([https://www.mardi4nfdi.de/about/task-areas MaRDI]) and will serve as an example of how to fairly compare different implementations of algorithms across a standardized benchmark set for scientific computing.

== Overview ==
An overview of MaRDI and MORB are provided in the poster, which was presented at [https://more.sciencesconf.org/ Model Reduction and Surrogate Modeling (MORE)], 19-23 September 2022, Berlin, Germany.

[[File:Morb2022.pdf|180px]]

== mini-MORB (snapshot of MATLAB prototype) ==
A MATLAB prototype for linear-time-invariant, first-order systems (LTI-FOS) is largely operational. An example [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/File:CdPlayer_n120m2q2.pdf report] for the [[CD Player]] benchmark is provided.

Please also see the [https://onlinelibrary.wiley.com/doi/10.1002/pamm.202300147 report] published in PAMM, as well as the accompanying [https://zenodo.org/record/8093833 code] hosted at Zenodo.

<nowiki>
@article{BenLS23,
title = {Towards a benchmark framework for model order reduction in the {{Mathematical Research Data Initiative}} ({{MaRDI}})},
author = {Benner, Peter and Lund, Kathryn and Saak, Jens},
year = {2023},
journal = {Proceeding Appl. Math. Mech.},
doi = {10.1002/pamm.202300147},
pages = {e202300147}
}
@misc{mini-MORB,
title = {mini-{{MORB}} ({{Model Order Reduction Benchmarker}})},
author = {Lund, Kathryn and Saak, Jens and Benner, Peter},
year = {2023},
doi = {10.5281/zenodo.8093832}
}
</nowiki>

FEniCS Rail

2023-09-12T14:16:03Z

Lund: Initialized fenicsRail page

{{preliminary}} 

[[Steel Profile|Oberwolfach Rail]] Reimplementation in FEniCS.

See the [https://gitlab.mpi-magdeburg.mpg.de/models/fenicsrail Gitlab] or [https://zenodo.org/record/5113560 Zenodo] links for the data.

MORB

2023-07-06T08:40:10Z

Lund: explain MORB acronym

MORB

2023-07-06T08:39:19Z

Lund: add link to code and arXiv

A new benchmarking tool for model order reduction is currently under development. It is a demonstrator of the generic benchmark framework developed for Task Area 2 of the Mathematical Research Data Initiative ([https://www.mardi4nfdi.de/about/task-areas MaRDI]) and will serve as an example of how to fairly compare different implementations of algorithms across a standardized benchmark set for scientific computing.

== Overview ==
An overview of MaRDI and MORB are provided in the poster, which was presented at [https://more.sciencesconf.org/ Model Reduction and Surrogate Modeling (MORE)], 19-23 September 2022, Berlin, Germany.

[[File:Morb2022.pdf|180px]]

== mini-MORB (snapshot of MATLAB prototype) ==
A MATLAB prototype for linear-time-invariant (LTI) systems is largely operational. An example [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/File:CdPlayer_n120m2q2.pdf report] for the [[CD Player]] benchmark is provided. Code can be found on [https://zenodo.org/record/8093833 Zenodo] and an accompanying report on [https://arxiv.org/abs/2307.00137 arXiv].

Search Tool

2023-05-11T12:26:38Z

Lund: add link to Glossary

{{preliminary}} 

We are currently developing a search interface that will allow users to filter benchmark data sets based on properties like size, linearity, symmetry, etc., similar to how one can navigate the SuiteSparse[https://sparse.tamu.edu] matrix collection. Please check back often for updates.

See the [[Glossary]] for a list of definitions, abbreviations, and search properties.

Main Page

2023-05-11T12:25:32Z

Lund: /* How to use the MOR Wiki */ add link to glossary

<Big>'''Welcome to the MOR Wiki'''</Big>

==Introduction==

<div class="thumb tright" style="width:302px;height:100vh">
<wikitwidget class="twitter-timeline" href="https://twitter.com/mor_wiki" data-widget-id="428887480843001856"/>
</div>

The purpose of the '''M'''odel-'''O'''rder-'''R'''eduction-'''Wiki''' is to bring together experts in the area of model reduction along with researchers from related application areas, with the goal of providing a platform for exchanging ideas and benchmark examples.

Modeling and numerical simulation are unavoidable in many application and research areas such as reaction processes, micro-electro-mechanical systems ([[:Wikipedia:MEMS|MEMS]]) design, and control design.
Such processes or devices can be modeled by partial differential equations ([[:Wikipedia:Partial_differential_equation|PDEs]]).
To simulate these models, spatial discretization via, for example, finite element discretization is necessary, which results in a system of ordinary differential equations ([[:Wikipedia:Ordinary_differential_equation|ODEs]]), or differential algebraic equations ([[:Wikipedia:Differential_algebraic_equation|DAEs]]).

After spatial discretization, the number of degrees of freedom ([[:Wikipedia:Degrees_of_Freedom|DoFs]]) is usually very high.
Simulating such large-scale systems of ODEs or DAEs can therefore become incredibly time-consuming.
Developed from well established mathematical theory and robust numerical algorithms, Model Order Reduction (MOR) or Model Reduction has been recognized as very efficient for reducing the simulation time of large-scale systems; see the page [[Projection based MOR]] for a basic overview.
Through model order reduction, a small system with a reduced number of equations (i.e., the reduced model) is derived.
The reduced model is simulated instead, and the solution of the original PDEs or ODEs can then be recovered from the solution of the reduced model.
As a result, the simulation time of the original large-scale system can be shortened by several orders of magnitude.
The reduced model as a whole can also replace the original system and be reused repeatedly during the design process, which can lead to further time or energy savings.

[[:Category:Parametric|Parametric model order reduction (PMOR) methods]] are designed for model order reduction of parametrized systems, where the parameters of the system play an important role in practical applications such as [[:Wikipedia:Integrated circuit design|Integrated Circuit (IC)]] design, [[:Wikipedia:MEMS|MEMS]] design, and chemical engineering.
The parameters could be the variables describing geometrical measurements, material properties, the damping of the system or the component flow-rate.
Reduced models are constructed such that all the parameters can be preserved with acceptable accuracy.

=== How to use the MOR Wiki ===
The MOR Wiki comprises pages providing [[:Category:Benchmark|benchmarks of parametric or non-parametric models]] and pages explaining applicable (P)MOR [[:Category:Method|methods]] as well as available [[:Category:Software|software implementations]]. We ([[MOR Wiki:About|the editors]]) are actively improving documentation at the moment, so check back often for updates!

Following the [[Submission rules]], it is also possible to submit new benchmarks, methods, or software packages, along with pages describing the contribution.

Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using Wiki software and the Wiki markup language we use here.

See the [[Glossary]] for definitions and abbreviations.

Find [[MOR_Wiki:Current_events|current model reduction conferences, workshops and minisymposia]], a [[Publications|list of books and lectures on model reduction]] and [[MOR_Wiki:Community_portal|links to the model reduction community]].

=== MORB: Model Order Reduction Benchmarker ===
A new benchmarking tool for model order reduction is currently under development. It is a demonstrator of the generic benchmark framework developed for Task Area 2 of the Mathematical Research Data Initiative ([https://www.mardi4nfdi.de/about/task-areas MaRDI] and will serve as an example of how to fairly compare different implementations of algorithms across a standardized benchmark set for scientific computing. Visit the [[MORB]] page for updates.

=== Citations ===
To cite the MOR Wiki itself, please use the following

The MORwiki Community. '''MORwiki - Model Order Reduction Wiki'''. http://modelreduction.org

<nowiki>
@misc{morwiki,
author = {{The MORwiki Community}},
title = {{MORwiki} - {M}odel {O}rder {R}eduction {W}iki},
howpublished = {\url{http://modelreduction.org}},
key = {morwiki}
}
</nowiki>

A BibTeX file which contains a list of references related to model order reduction can be found and downloaded here: [[Bibtex|mor.bib]].

== List of all Categories in the Wiki ==
{{special:categories}}

{|
|- valign="top"
|<categorytree mode=all>Benchmark</categorytree> || <categorytree mode=all>Method</categorytree> || <categorytree mode=all>Software</categorytree> || <categorytree mode=all>Miscellaneous</categorytree>
|}

Nonlinear RC Ladder

2023-05-11T12:24:17Z

Lund: remove link

[[Category:benchmark]]
[[Category:procedural]]
[[Category:nonlinear]]
[[Category:SISO]]

==Description==
<figure id="fig:nrcl">[[File:nrcl.png|400px|thumb|right| Nonlinear RC-Ladder]]</figure>

The nonlinear RC-ladder is an electronic test circuit first introduced in <ref name="chen99"/>, and its variant is also introduced in <ref name="RewW03"/>.
These nonlinear first-order system model a resistor-capacitor network that exhibits a distinct nonlinear behaviour caused by either the nonlinear resistors consisting of a parallel connected resistor with a diode (see the right figure) or the nonlinear resistors connected parallel to the capacitor (see Fig. 7 in <ref name = "RewW03"/>).

===Model 1===
First, we discuss the modelling of an RC circuit, where the nonlinear resistors consist of a parallel connected resistor with a diode as shown in the above figure. For this, the underlying model is given by a SISO gradient system of the form <ref name="condon04"/>:

:<math>
\dot{x}(t) = \begin{pmatrix} -g(x_1(t)) - g(x_1(t) - x_2(t)) \\ g(x_1(t)-x_2(t)) - g(x_2(t)-x_3(t)) \\ \vdots \\ g(x_{k-1}(t) - x_k(t)) - g(x_k(t) - x_{x+1}(t)) \\ \vdots \\ g(x_{N-1}(t) - x_N(t)) \end{pmatrix}+\begin{pmatrix}u(t) \\ 0 \\ \vdots \\ 0 \\ \vdots \\ 0 \end{pmatrix},
</math>

:<math>
y(t) = x_1(t),
</math>

where the <math>g</math> is a mapping <math>g(x_i):\mathbb{R} \to \mathbb{R}</math>:

:<math>
g(x_i) = g_D(x_i) + x_i,
</math>

which combines the effect of a [[wikipedia:Diode|diode]] and a resistor.

====Nonlinearity====
The nonlinearity <math>g_D</math> models a diode as a nonlinear resistor,
based on the [[wikipedia:Diode_modelling#Shockley_diode_model|Shockley model]] <ref name="reis14"/>:

:<math>
g_D(x_i) = i_S (\exp(u_P x_i) - 1),
</math>

with material parameters <math>i_S > 0</math> and <math>u_P > 0</math>.

For this benchmark the parameters are selected as: <math>i_S = 1</math> and <math>u_P = 40</math> as in <ref name="chen99"/>.

===Model 2===
Second, we discuss the modelling of an RC circuit, where the nonlinear resistors are connected parallel to the capacitors (see Fig. 7 in <ref name = "RewW03"/>). For this, the underlying model is also given by a SISO gradient system of the form <ref name="RewW03"/>:

:<math>
\dot{x}(t) = \begin{pmatrix} -2 & 1& & \\ 1 & -2 & 1 & \\ & \ddots & \ddots & \ddots\\ & & 1 & -2 \end{pmatrix} x(t) + g(x),
</math>

and the output function:

:<math>
y(t) = x_1(t),
</math>

where the <math>g</math> is a mapping <math>g(x_i):\mathbb{R} \to \mathbb{R}</math>, respresenting the effect of a nonlinear resistor.

:<math>
g(x_i) = a\cdot \text{sgn}(x_i)\cdot x_i^2
</math>

which represents the effect of a nonlinear resistor, and sgn denotes the [[wikipedia:Sign_function|sign function]].

====Nonlinearity====
The nonlinearity <math>g_D</math> models a diode as a nonlinear resistor <ref name="RewW03"/>:

:<math>
g(x_i) = a\cdot \text{sgn}(x_i)\cdot x_i^2
</math>

which represents the effect of a nonlinear resistor, and sgn denotes the [[wikipedia:Sign_function|sign function]].

For this benchmark the parameters are selected as: <math>a = 1</math> as in <ref name="chen99"/>.

Alternatively, the variant from <ref name="kawano19"/> can be used, featuring the nonlinearity:

:<math>
g(x_i) = \frac{x_i^2}{2} + \frac{x_i^3}{3}.
</math>

This represents practically a resistor-inductor cascade with nonlinear resistors.

===Model 3===
Third, a circuit of chained [[wikipedia:Inverter_(logic_gate)|inverter gates]], a so-called inverter chain <ref name="gu12"/>, is presented.
This is a SISO system, given by:

:<math>
\dot{x}(t) = -x(t) + \begin{pmatrix} 0 \\ g(x_1(t)) \\ \vdots \\ g(x_{N-1}(t)) \end{pmatrix} + \begin{pmatrix} u(t) \\ 0 \\ \vdots \\ 0 \end{pmatrix},
</math>

and the output function:

:<math>
y(t) = x_N(t).
</math>

====Nonlinearity====

The nonlinear function <math>g</math> is a mapping <math>g(x_i):\mathbb{R} \to \mathbb{R}</math> describing the inverter characteristic:

:<math>
g(x_i) = v \tanh(a x_i),
</math>

being a parameterized [[wikipedia:Hyperbolic_function|hyperbolic tangent]] with the supply voltage <math>v=1</math> and a physical parameter <math>a</math>, i.e. <math>a=5</math>.

===Input===

As an external input, several alternatives are presented in <ref name="chen00"/>, which are listed next.
A simple step function is given by:
:<math>
u_1(t)=\begin{cases}0 & t < 4 \\ 1 & t \geq 4 \end{cases},
</math>

an exponential decaying input is provided by:
:<math>
u_2(t) = e^{-t}.
</math>

Additional input sources are given by conjunction of sine waves with different periods <ref name="condon04a"/>:
:<math>
u_3(t) = \sin(2\pi 50t)+\sin(2\pi 1000t),
</math>

:<math>
u_4(t) = \sin(2\pi 50t) \sin(2\pi 1000t).
</math>

==Data==

A sample procedural MATLAB implementation for order <math>N</math> for all three models is given by:

<div class="thumbinner" style="width:540px;text-align:left;">
<source lang="matlab">

function [f,B,C] = nrc(N,model)
%% Procedural generation of "Nonlinear RC Ladder" benchmark system

B = sparse(1,1,1,N,1); % input matrix
C = sparse(1,1,1,1,N); % output matrix

switch lower(model)

case 'shockley'

g = @(x) exp(40.0*x) + x - 1.0;

A0 = sparse(1,1,1,N,N);

A1 = spdiags(ones(N-1,1),-1,N,N) - speye(N);
A1(1,1) = 0;

A2 = spdiags([ones(N-1,1);0],0,N,N) - spdiags(ones(N,1),1,N,N);

f = @(x) -g(A0*x) + g(A1*x) - g(A2*x);

case 'sign'

A = gallery('tridiag',N,1,-2,1);

f = @(x) A*x - sign(x).*(x.*x);

case 'ind'

A = gallery('tridiag',N,1,-2,1);

f = @(x) A*x - ((x.^2)./2 + (x.^3)./3);

case 'inv'

f = @(x) [0;tanh(x(2:end))] - x;

C = sparse(1,N,1,1,N);

otherwise

error('Choose shockley, sign, ind or inv');
end
end

</source>
</div>

Here the vector field is realized in a vectorized form as a [[wikipedia:Closure_(computer_programming)|closure]].

==Dimensions==

System structure:
:<math>
\begin{align}
\dot{x}(t) &= f(x(t)) + Bu(t), \\
y(t) &= Cx(t).
\end{align}
</math>

System dimensions:

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

==Citation==

To cite these benchmarks, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Nonlinear RC Ladder'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Nonlinear_RC_Ladder

@MISC{morwiki_modNonRCL,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Nonlinear RC Ladder},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Nonlinear_RC_Ladder}</nowiki>,
year = 2018
}

* For the background on the benchmarks:
** for Model 1 [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morChe99 morChe99] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morChe99 BibTeX])
** for Model 2 [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morRew03 morRew03] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morRew03 BibTeX])
** for Model 3 [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morGu12 morGu12] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morGu12 BibTeX])

==References==

<references>

<ref name="chen99">Y. Chen, "[http://hdl.handle.net/1721.1/9381 Model Reduction for Nonlinear Systems]", Master Thesis, 1999.</ref>

<ref name="RewW03">M. Rewienski and J. White, "[https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1174092 A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices]", IEEE Transactions on computer-aided design of integrated circuits and systems 22(2): 155--170, 2003.</ref>

<ref name="chen00">Y. Chen and J. White, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.19.8951&rep=rep1&type=pdf A Quadratic Method for Nonlinear Model Order Reduction]", Int. Conference on Modelling and Simulation of Microsystems Semiconductors, Sensors and Actuators, 2000.</ref>

<ref name="condon04">M. Condon and R. Ivanov, "[https://doi.org/10.1007/s00332-004-0617-5 Empirical Balanced Truncation for Nonlinear Systems]", Journal of Nonlinear Science 14(5):405--414, 2004.</ref>

<ref name="condon04a">M. Condon and R. Ivanov, "[https://doi.org/10.1108/03321640410510730 Model Reduction of Nonlinear Systems]", COMPEL 23(2): 547--557, 2004</ref>

<ref name="gu12">C. Gu, "[https://www2.eecs.berkeley.edu/Pubs/TechRpts/2012/EECS-2012-217.pdf Model Order Reduction of Nonlinear Dynamical Systems]", PhD Thesis (University of California, Berkeley), 2012.</ref>

<ref name="reis14">T. Reis. "[https://doi.org/10.1007/978-3-319-08437-4_2 Mathematical Modeling and Analysis of Nonlinear Time-Invariant RLC Circuits]", In: Large-Scale Networks in Engineering and Life Sciences. Modeling and Simulation in Science, Engineering and Technology: 125--198, 2014.</ref>

<ref name="kawano19">Y. Kawano, J.M.A. Scherpen. "[https://arxiv.org/abs/1902.09836 Empirical Differential Gramians for Nonlinear Model Reduction]", arXiv (cs.SY): 1902.09836, 2019.</ref>

</references>

==Contact==

''[[User:Himpe|Christian Himpe]]''

Projection based MOR

2023-05-11T12:24:01Z

Lund: remove link

[[Category:method]]
[[Category:time invariant]]
Consider the linear time invariant system

:<math>
E \frac{dx(t)}{dt}=A x(t)+B u(t), \quad
y(t)=Cx(t), \quad \quad (1)
</math>

as an example.
All the existing model order reduction (MOR) methods are based on projection<ref>Antoulas, A. C. "[http://dx.doi.org/10.1137/1.9780898718713 Approximation of large-scale dynamical systems]". Vol. 6. Society for Industrial and Applied Mathematics, pp.376--377, 2009; ISBN 978-0-89871-529-3</ref>.
That is to find a subspace <math>S_1</math> which approximates the manifold where the state vector <math>x(t)</math> resides.
Afterwards, <math>x(t)</math> is approximated by its projection <math>\hat x(t)</math> in <math>S_1</math>.
The reduced model is produced by [[wikipedia:Petrov–Galerkin_method|Petrov-Galerkin projection]] using a test subspace <math>S_2</math>, or by [[wikipedia:Galerkin_method|Galerkin projection]] using <math>S_1</math> as the test subspace.
Assuming that an orthonormal basis <math>V=(v_1,v_2, \ldots, v_q)</math> of the subspace <math>S_1</math> has been found, then the approximation <math>\hat x(t)</math> in <math>S_1</math> can be represented by the basis as <math>\hat{x} (t)=V z(t)</math>.
Therefore <math>x(t)</math> can be approximated by <math> x(t) \approx V z(t)</math>.
Here <math>z</math> is a vector of length <math>q \ll n</math>.
Once <math>z(t)</math> is computed, an approximate solution <math>\hat x(t)=V z(t)</math> for <math>x(t)</math> can be obtained.
The vector <math>z(t)</math> can be computed from the reduced model, derived by the following two steps.

Step 1. By replacing <math>x</math> in (1) with <math>Vz</math>, we get

:<math>E \frac{d{Vz}}{dt}=A Vz+Bu(t)+e,\quad \hat y(t)=CV z.</math>

Step 2. The residual is denoted as <math>e=AVz+Bu(t)-E \frac{d{Vz}}{dt}</math>. Forcing <math>e=0</math> in a properly chosen subspace <math>S_2</math> of <math>\mathbb {R}^n</math> leads to the Petrov-Galerkin projection: <math>W^T e=0</math>, where the columns of <math>W</math> are the basis of <math>S_2</math>.
Then we have,

:<math>W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t), \quad \hat{y}(t)=CVz.</math>

By defining <math>\hat{E}=W^TEV</math>, <math>\hat {A}=W^TAV, \hat{B}=W^TB</math>, <math>\hat{C}=CV</math>, we get the final reduced model

:<math>\hat{E} \frac{dz(t)}{dt}=\hat{A}z(t)+\hat{B}u(t), \quad
\hat{y}(t)=\hat{C}z(t). \quad \quad (2)
</math>

Notice that the approximation <math>\hat x(t)=Vz(t)</math> of <math>x(t)</math> can be obtained from <math> z(t)</math> by solving the system in (2).
The system in (2) is much smaller than the system in (1) in the sense that there are many less equations in (2) than in (1).
Therefore, the system in (2) is much easier to be solved, which is the so-called reduced model.
In order to save the simulation time of solving (1), the reduced model can be used to replace (1) to attain a fast simulation. Furthermore, the error between the two systems should be within acceptable tolerance.
The error can be measured through the error between the the state vectors <math>x(t), \hat x(t)</math>, or between the output responses <math>y(t), \hat y(t)</math>, or between the transfer functions of the two systems.
It can be seen that once the two matrices <math>W</math> and <math>V</math> have been computed, the reduced model is obtained.
While the Gramian based MOR methods (e.g. [[Balanced Truncation]]) usually compute <math>W</math> different from <math>V</math>, some methods use <math>W=V</math>, e.g. some of the [[Moment-matching method]]s, the [[Reduced Basis PMOR method]]s, and some of the [[POD method]]s etc.. When <math>W=V</math>, Petrov-Galerkin projection becomes Galerkin projection.
MOR methods differ in the computation of the two matrices <math>W</math> and <math>V</math>.
The Gramian based MOR methods compute <math>W</math> and <math>V</math> by the controllability and observability Gramians.
Reduced basis methods and POD methods compute <math>V</math> from the snapshots of the state vector <math>x(t)</math> at different time steps (also at the samples of the parameters if the system is parametric). The methods based on moment-matching compute :<math>W</math> and <math>V</math> from the moments of the transfer function.
In eigenvalue based MOR methods, e.g. [[Modal truncation]], the columns of :<math>W</math> and <math>V</math> are eigenvectors or invariant subspaces corresponding to selected eigenvalues of the matrix pair <math>(A,E)</math>.

One common goal of all MOR methods is that the behavior of the reduced model should be sufficiently "close" to that of the original model guaranteed through the above mentioned error measurements.

==References==

<references/>

Branchline Coupler

2023-05-11T12:23:33Z

Lund: remove link

{{preliminary}} 

[[Category:benchmark]]
[[Category:Parametric]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:second differential order]]

==Description==

A '''branchline coupler''' (see Fig. 1) is a microwave semiconductor device, which is simulated by the [http://www.maxwells-equations.com/forms.php#harmonic time-harmonic Maxwell's equation].
A 2-section '''branchline coupler''' consists of four strip line ports, coupled to each other by two transversal bridges.
The energy excited at one port is coupled almost in equal shares to the two opposite ports, when considered as a MIMO-system.
Here, only the SISO case is considered.
The '''branchline coupler''' with <math>0.05mm</math> thickness is placed on a substrate with <math>0.749mm</math> thickness and relative permittivity
<math> \epsilon_r = 2.2 </math> and zero-conductivity <math> \sigma = 0 S/m </math>.
The simulation domain is confined to a <math> 23.6 \times 22 \times 7 mm^3 </math> box.
The metallic ground plane of the device is represented by the electric boundary condition. The magnetic boundary
condition is considered for the other sides of the structures. The discrete input port with source impedance <math>50 \Omega</math>
imposes <math>1 A</math> current as the input. The voltage along the coupled port at the end of the other side of the coupler is
read as the output.

<figure id="fig:branch">
[[File:BranchlineCoupler.png|frame|<caption>Branchline Coupler Model<ref name="hess13"/></caption>]]
</figure>

Considered parameters are the frequency <math>\omega </math> and the relative permeability <math> \mu_r </math> .

The affine form <math> a(u, v; \omega, \mu_r) = \sum_{q=1}^Q \Theta^q(\omega, \mu_r) a^q(u, v) </math> can be established using <math> Q = 2 </math> affine terms.

The discretized bilinear form is <math> a(u, v; \omega, \mu_r) = \sum_{q=1}^Q \Theta^q(\omega, \mu_r) A^q </math>, with matrices <math> A^q </math>.

The matrices corresponding to the bilinear forms <math> a^q( \cdot , \cdot ) </math> as well as the input and output forms and the H(curl) inner product matrix have been assembled
using the [[wikipedia:Finite_Element_Method|Finite Element Method]], resulting in <math>27679</math> degrees of freedom, after removal of boundary conditions.

The coefficient functions are given by:

:<math> \Theta^1(\omega, \mu_r) = \frac{1}{\mu_r} </math>

:<math> \Theta^2(\omega, \mu_r) = -\omega^2. </math>

The parameter domain of interest is <math> \omega \in [1.0, 10.0] * 10^9 Hz</math>, where the factor of <math> 10^9 </math> has already been taken into account
while assembling the matrices, while the material variation occurs between <math> \mu_r \in [0.5, 2.0] </math>. The input functional also has a factor of <math> \omega </math>.

==Data==

The files are numbered according to their appearance in the summation and can be found here:
Part1
[[Media:branchline_part1.zip|branchline_part1.zip]]
Part2
[[Media:branchline_part2.zip|branchline_part2.zip]]
Part3
[[Media:branchline_part3.zip|branchline_part3.zip]]

==Origin==

The models have been developed within the [http://www.moresim4nano.org MoreSim4Nano project].

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_branchcouple,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Branchline Coupler},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Branchline_Coupler}</nowiki>,
year = {2013}
}

* For the background on the benchmark:
@ARTICLE{morHesB13,
author = {M.~W. Hess and P. Benner},
title = {Fast Evaluation of Time-Harmonic {M}axwell's
Equations Using the Reduced Basis Method},
journal = {{IEEE} Trans. Microw. Theory Techn.},
year = 2013,
volume = 61,
number = 6,
pages = {2265--2274},
doi = {10.1109/TMTT.2013.2258167}
}

==References==

<references>

<ref name="hess13">M. W. Hess, P. Benner, "[https://doi.org/10.1109/TMTT.2013.2258167 Fast Evaluation of Time-Harmonic Maxwell's Equations Using the Reduced Basis Method]", IEEE Transactions on Microwave Theory and Techniques, 61(6): 2265--2274, 2013.</ref>

</references>

==Contact==

'' [[User:hessm|Martin Hess]]''

PyMOR

2023-05-11T12:22:57Z

Lund: remove link to list of abbrev

[[Category:Software]]
[[Category:Python]]

== Synopsis ==

[https://pymor.org pyMOR] is a [https://opensource.org/licenses/BSD-2-Clause BSD-licensed] software library for building model order reduction applications in the [[:Wikipedia:Python_(programming_language)|Python programming language]].
Implemented algorithms include reduced basis methods for parametric linear and non-linear problems, as well as system-theoretic methods such as balanced truncation and iterative rational Krylov algorithm.
'''pyMOR''' is designed from the ground up for easy integration with external PDE solver packages but also offers Python-based discretizations for getting started easily.

== Features ==

Currently, the following model reduction algorithms are provided by '''pyMOR''':
* A generic reduction routine for projection of arbitrary high-dimensional discretizations onto reduced spaces, preserving (possibly nested) affine decompositions of operators and functionals for efficient offline/online decomposition.
* Efficient error estimation for linear affinely decomposed problems.
* Empirical interpolation of arbitrary operators (with efficient evaluation of projected interpolated operators if the operator supports restriction to selected degrees of freedom).
* Parallel adaptive greedy and POD algorithms for reduced space construction.
* Empirical-Interpolation-Greedy and DEIM algorithms for generation of interpolation data for empirical operator interpolation.
* Balanced-based and interpolation-based reduction methods for first-order and second-order linear time-invariant systems.
* Model order reduction using artificial neural networks.
* Data-driven model order reduction with Dynamic Mode Decomposition, Eigensystem Realization, parametric AAA.
* [[:Wikipedia:Gram_schmidt|Gram-Schmidt algorithm]] supporting re-orthogonalization for improved numerical accuracy.
* Time-stepping and Newton algorithms, as well as generic iterative linear solvers.
* Low-rank alternating direction implicit (LR ADI) method for large-scale Lyapunov equations and bindings for matric equations solvers in [http://slicot.org SLICOT] (via [https://github.com/python-control/Slycot slycot]) and [https://www.mpi-magdeburg.mpg.de/projects/mess Py-M.E.S.S].
* Eigenvalue/pole computation using the implicitly restarted Arnoldi method and the subspace accelerated dominant pole (SAMDP) algorithm.
* Modal truncation for linear time-invariant systems.
* Time-dependent parameters.

All these algorithms are formulated in terms of abstract interfaces for seamless integration with external high-dimensional PDE solvers.
Bindings for the following PDE solver libraries are available:
* [http://www.dealii.org/ deal.II]
* [http://dune-project.org/ DUNE]
* [http://fenicsproject.org/ FEniCS]
* [http://sourceforge.net/projects/ngsolve/ NGSolve]
* [https://scikit-fem.readthedocs.io/ scikit-fem] (experimental)

Pure Python implementations of discretizations using the [https://www.scipy.org NumPy/SciPy] scientific computing stack are implemented to provide an easy-to-use sandbox for experimentation with new model reduction approaches. '''pyMOR''' offers:
* Structured 1D and 2D grids, as well as an experimental Gmsh-based grid, implementing the same abstract grid interface.
* [[:Wikipedia:Finite_element|Finite element]] and [[:Wikipedia:Finite_volume|finite volume]] operators based on this interface.
* SciPy/[http://crd-legacy.lbl.gov/~xiaoye/SuperLU SuperLU] based iterative and direct solvers for sparse systems.
* [[:Wikipedia:Opengl|OpenGL]] and [http://matplotlib.org matplotlib] based visualizations of solutions.

== References ==
* M. Ohlberger, S. Rave, S. Schmidt, S. Zhang. "[http://dx.doi.org/10.1007/978-3-319-05591-6_69 A Model Reduction Framework for Efficient Simulation of Li-Ion Batteries]". Springer Proceedings in Mathematics & Statistics Vol. 78: Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, Berlin, June 2014.
* R. Milk, S. Rave, F. Schindler. "[https://doi.org/10.1137/15M1026614 pyMOR - Generic Algorithms and Interfaces for Model Order Reduction]". SIAM Journal on Scientific Computing 38(5): S194--S216, 2016.

== Links ==
* Official [https://pymor.org website],
* Development of '''pyMOR''' can be tracked on [https://github.com/pymor/pymor GitHub],
* Online [https://docs.pymor.org documentation].

== Contact ==
For assistance with, and contributions to '''pyMOR''', the developers can be contacted via [https://github.com/pymor/pymor/discussions GitHub discussions].

Nonlinear RC Ladder

2023-05-11T12:21:56Z

Lund: remove link to list of abbreviations

[[Category:benchmark]]
[[Category:procedural]]
[[Category:nonlinear]]
[[Category:SISO]]

==Description==
<figure id="fig:nrcl">[[File:nrcl.png|400px|thumb|right| Nonlinear RC-Ladder]]</figure>

The nonlinear RC-ladder is an electronic test circuit first introduced in <ref name="chen99"/>, and its variant is also introduced in <ref name="RewW03"/>.
These nonlinear first-order system model a resistor-capacitor network that exhibits a distinct nonlinear behaviour caused by either the nonlinear resistors consisting of a parallel connected resistor with a diode (see the right figure) or the nonlinear resistors connected parallel to the capacitor (see Fig. 7 in <ref name = "RewW03"/>).

===Model 1===
First, we discuss the modelling of an RC circuit, where the nonlinear resistors consist of a parallel connected resistor with a diode as shown in the above figure. For this, the underlying model is given by a SISO gradient system of the form <ref name="condon04"/>:

:<math>
\dot{x}(t) = \begin{pmatrix} -g(x_1(t)) - g(x_1(t) - x_2(t)) \\ g(x_1(t)-x_2(t)) - g(x_2(t)-x_3(t)) \\ \vdots \\ g(x_{k-1}(t) - x_k(t)) - g(x_k(t) - x_{x+1}(t)) \\ \vdots \\ g(x_{N-1}(t) - x_N(t)) \end{pmatrix}+\begin{pmatrix}u(t) \\ 0 \\ \vdots \\ 0 \\ \vdots \\ 0 \end{pmatrix},
</math>

:<math>
y(t) = x_1(t),
</math>

where the <math>g</math> is a mapping <math>g(x_i):\mathbb{R} \to \mathbb{R}</math>:

:<math>
g(x_i) = g_D(x_i) + x_i,
</math>

which combines the effect of a [[wikipedia:Diode|diode]] and a resistor.

====Nonlinearity====
The nonlinearity <math>g_D</math> models a diode as a nonlinear resistor,
based on the [[wikipedia:Diode_modelling#Shockley_diode_model|Shockley model]] <ref name="reis14"/>:

:<math>
g_D(x_i) = i_S (\exp(u_P x_i) - 1),
</math>

with material parameters <math>i_S > 0</math> and <math>u_P > 0</math>.

For this benchmark the parameters are selected as: <math>i_S = 1</math> and <math>u_P = 40</math> as in <ref name="chen99"/>.

===Model 2===
Second, we discuss the modelling of an RC circuit, where the nonlinear resistors are connected parallel to the capacitors (see Fig. 7 in <ref name = "RewW03"/>). For this, the underlying model is also given by a ([[List_of_abbreviations#SISO|SISO]]) gradient system of the form <ref name="RewW03"/>:

:<math>
\dot{x}(t) = \begin{pmatrix} -2 & 1& & \\ 1 & -2 & 1 & \\ & \ddots & \ddots & \ddots\\ & & 1 & -2 \end{pmatrix} x(t) + g(x),
</math>

and the output function:

:<math>
y(t) = x_1(t),
</math>

where the <math>g</math> is a mapping <math>g(x_i):\mathbb{R} \to \mathbb{R}</math>, respresenting the effect of a nonlinear resistor.

:<math>
g(x_i) = a\cdot \text{sgn}(x_i)\cdot x_i^2
</math>

which represents the effect of a nonlinear resistor, and sgn denotes the [[wikipedia:Sign_function|sign function]].

====Nonlinearity====
The nonlinearity <math>g_D</math> models a diode as a nonlinear resistor <ref name="RewW03"/>:

:<math>
g(x_i) = a\cdot \text{sgn}(x_i)\cdot x_i^2
</math>

which represents the effect of a nonlinear resistor, and sgn denotes the [[wikipedia:Sign_function|sign function]].

For this benchmark the parameters are selected as: <math>a = 1</math> as in <ref name="chen99"/>.

Alternatively, the variant from <ref name="kawano19"/> can be used, featuring the nonlinearity:

:<math>
g(x_i) = \frac{x_i^2}{2} + \frac{x_i^3}{3}.
</math>

This represents practically a resistor-inductor cascade with nonlinear resistors.

===Model 3===
Third, a circuit of chained [[wikipedia:Inverter_(logic_gate)|inverter gates]], a so-called inverter chain <ref name="gu12"/>, is presented.
This is a SISO system, given by:

:<math>
\dot{x}(t) = -x(t) + \begin{pmatrix} 0 \\ g(x_1(t)) \\ \vdots \\ g(x_{N-1}(t)) \end{pmatrix} + \begin{pmatrix} u(t) \\ 0 \\ \vdots \\ 0 \end{pmatrix},
</math>

and the output function:

:<math>
y(t) = x_N(t).
</math>

====Nonlinearity====

The nonlinear function <math>g</math> is a mapping <math>g(x_i):\mathbb{R} \to \mathbb{R}</math> describing the inverter characteristic:

:<math>
g(x_i) = v \tanh(a x_i),
</math>

being a parameterized [[wikipedia:Hyperbolic_function|hyperbolic tangent]] with the supply voltage <math>v=1</math> and a physical parameter <math>a</math>, i.e. <math>a=5</math>.

===Input===

As an external input, several alternatives are presented in <ref name="chen00"/>, which are listed next.
A simple step function is given by:
:<math>
u_1(t)=\begin{cases}0 & t < 4 \\ 1 & t \geq 4 \end{cases},
</math>

an exponential decaying input is provided by:
:<math>
u_2(t) = e^{-t}.
</math>

Additional input sources are given by conjunction of sine waves with different periods <ref name="condon04a"/>:
:<math>
u_3(t) = \sin(2\pi 50t)+\sin(2\pi 1000t),
</math>

:<math>
u_4(t) = \sin(2\pi 50t) \sin(2\pi 1000t).
</math>

==Data==

A sample procedural MATLAB implementation for order <math>N</math> for all three models is given by:

<div class="thumbinner" style="width:540px;text-align:left;">
<source lang="matlab">

function [f,B,C] = nrc(N,model)
%% Procedural generation of "Nonlinear RC Ladder" benchmark system

B = sparse(1,1,1,N,1); % input matrix
C = sparse(1,1,1,1,N); % output matrix

switch lower(model)

case 'shockley'

g = @(x) exp(40.0*x) + x - 1.0;

A0 = sparse(1,1,1,N,N);

A1 = spdiags(ones(N-1,1),-1,N,N) - speye(N);
A1(1,1) = 0;

A2 = spdiags([ones(N-1,1);0],0,N,N) - spdiags(ones(N,1),1,N,N);

f = @(x) -g(A0*x) + g(A1*x) - g(A2*x);

case 'sign'

A = gallery('tridiag',N,1,-2,1);

f = @(x) A*x - sign(x).*(x.*x);

case 'ind'

A = gallery('tridiag',N,1,-2,1);

f = @(x) A*x - ((x.^2)./2 + (x.^3)./3);

case 'inv'

f = @(x) [0;tanh(x(2:end))] - x;

C = sparse(1,N,1,1,N);

otherwise

error('Choose shockley, sign, ind or inv');
end
end

</source>
</div>

Here the vector field is realized in a vectorized form as a [[wikipedia:Closure_(computer_programming)|closure]].

==Dimensions==

System structure:
:<math>
\begin{align}
\dot{x}(t) &= f(x(t)) + Bu(t), \\
y(t) &= Cx(t).
\end{align}
</math>

System dimensions:

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

==Citation==

To cite these benchmarks, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Nonlinear RC Ladder'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Nonlinear_RC_Ladder

@MISC{morwiki_modNonRCL,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Nonlinear RC Ladder},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Nonlinear_RC_Ladder}</nowiki>,
year = 2018
}

* For the background on the benchmarks:
** for Model 1 [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morChe99 morChe99] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morChe99 BibTeX])
** for Model 2 [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morRew03 morRew03] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morRew03 BibTeX])
** for Model 3 [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morGu12 morGu12] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morGu12 BibTeX])

==References==

<references>

<ref name="chen99">Y. Chen, "[http://hdl.handle.net/1721.1/9381 Model Reduction for Nonlinear Systems]", Master Thesis, 1999.</ref>

<ref name="RewW03">M. Rewienski and J. White, "[https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1174092 A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices]", IEEE Transactions on computer-aided design of integrated circuits and systems 22(2): 155--170, 2003.</ref>

<ref name="chen00">Y. Chen and J. White, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.19.8951&rep=rep1&type=pdf A Quadratic Method for Nonlinear Model Order Reduction]", Int. Conference on Modelling and Simulation of Microsystems Semiconductors, Sensors and Actuators, 2000.</ref>

<ref name="condon04">M. Condon and R. Ivanov, "[https://doi.org/10.1007/s00332-004-0617-5 Empirical Balanced Truncation for Nonlinear Systems]", Journal of Nonlinear Science 14(5):405--414, 2004.</ref>

<ref name="condon04a">M. Condon and R. Ivanov, "[https://doi.org/10.1108/03321640410510730 Model Reduction of Nonlinear Systems]", COMPEL 23(2): 547--557, 2004</ref>

<ref name="gu12">C. Gu, "[https://www2.eecs.berkeley.edu/Pubs/TechRpts/2012/EECS-2012-217.pdf Model Order Reduction of Nonlinear Dynamical Systems]", PhD Thesis (University of California, Berkeley), 2012.</ref>

<ref name="reis14">T. Reis. "[https://doi.org/10.1007/978-3-319-08437-4_2 Mathematical Modeling and Analysis of Nonlinear Time-Invariant RLC Circuits]", In: Large-Scale Networks in Engineering and Life Sciences. Modeling and Simulation in Science, Engineering and Technology: 125--198, 2014.</ref>

<ref name="kawano19">Y. Kawano, J.M.A. Scherpen. "[https://arxiv.org/abs/1902.09836 Empirical Differential Gramians for Nonlinear Model Reduction]", arXiv (cs.SY): 1902.09836, 2019.</ref>

</references>

==Contact==

''[[User:Himpe|Christian Himpe]]''

Iterative Rational Krylov Algorithm

2023-05-11T12:21:27Z

Lund: remove link to list of abbrev

[[Category:method]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:linear algebra]]

==Description==

'''IRKA''' (iterative rational Krylov algorithm) is an interpolation-based model reduction method for SISO linear time invariant systems.

:<equation id="gensys" shownumber>
<math>
\dot{x}(t)=A x(t)+b u(t), \quad
y(t)=c^Tx(t),\quad A\in\mathbb{R}^{n\times n},~b\in\mathbb{R}^{n},~c\in\mathbb{R}^{n}.\qquad (1)
</math>
</equation>

For a given system <math>G </math> and a prescribed reduced system order <math>r</math>, the goal of the algorithm is to find a local minimizer <math>\hat{G} </math> for the <math> H_2 </math> model reduction problem

:<math>
||G-\hat{G} ||_{H_2} = \min_{\text{dim}(\tilde{G})=r} ||G-\tilde{G}||_{H_2}.
</math>

Initially investigated in <ref name="MeiL67"></ref>, first order necessary conditions for a local minimizer <math>\hat{G}</math> imply that its rational transfer function <math>\hat{G}(s)=\hat{c}^T (sI-\hat{A})^{-1}b</math> is a Hermite interpolant of the original transfer function at its reflected system poles, i.e.,

:<math>
G(-\hat{\lambda}_i) = \hat{G}(-\hat{\lambda}_i), \quad G'(-\hat{\lambda}_i) = \hat{G}'(-\hat{\lambda}_i), \quad, i =1,\dots,r,
</math>
where <math>\{\hat{\lambda}_1,\dots,\hat{\lambda}_r\} </math> are assumed to be the simple poles of <math> \hat{G} </math>. Based on the idea of rational interpolation by rational Krylov subspaces, in <ref name="GugAB08"></ref> the authors have picked up the optimality conditions and proposed to iteratively correct projection subspaces until interpolation at the reflected reduced system poles is ensured. In pseudocode, the classical '''IRKA''' algorithm from <ref name="GugAB08"></ref> looks like

1. Make an initial selection of <math>\sigma_i </math> for <math>i=1,\dots,r </math> that is closed under conjugation and fix a convergence tolerance <math>tol</math>.
2. Choose <math>V_r </math> and <math> W_r</math> so that <math>\text{Ran}(V_r) =\{(\sigma_1 I -A)^{-1}b,\dots,(\sigma_rI-A)^{-1}b \} </math>, <math>\text{Ran}(W_r) =\{(\sigma_1 I -A^T)^{-1}c,\dots, (\sigma_rI-A^T)^{-1}c \} </math> and <math> W_r^TV_r=I</math>.
3. while (relative change in <math>\{\sigma_i\} > tol</math>)
(a) <math>\hat{A} = W_r^TAV_r</math>
(b) Assign <math>\sigma_i \leftarrow -\lambda_i(\hat{A}),</math> for <math> i=1,\dots,r</math>
(c) Update <math>V_r</math> and <math>W_r</math> so that <math>\text{Ran}(V_r) =\{(\sigma_1 I -A)^{-1}b,\dots,(\sigma_rI-A)^{-1}b \} </math>, <math>\text{Ran}(W_r) =\{(\sigma_1 I -A^T)^{-1}c,\dots, (\sigma_rI-A^T)^{-1}c \} </math> and <math> W_r^TV_r=I</math>.
4. <math>\hat{A} = W_r^TAV_r, \hat{b}= W_r^Tb, \hat{c}^T = c^TV_r.</math>

Although a rigorous convergence proof so far has only be given for symmetric state space systems <ref name="FlaBG12"></ref>, numerous experiments have shown that the algorithm often converges rapidly. Moreover, the algorithm has been extended to, e.g., multiple-input-multiple output, discrete time and differential algebraic systems.

== A minimal example ==

For the lecture Model [http://www.itm.uni-stuttgart.de/courses/model_reduction/model_reduction_en.php MOR] of Mechanical System from the Institute of Engineering and Computational Mechanics University of Stuttgart a very simple example of the IRKA Algorithm were written.

The implementation with two basic exampless can be found here

[[Media:IRKA_Example.zip]]

This code is published under the BSD3-Clause License
All rights reserved. (c) 2015, Joerg.Fehr, University of Stuttgart

==References==
<references>

<ref name="MeiL67"> L. Meier, D.G. Luenberger, "[http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1098680&tag=1 Approximation of linear constant systems]", IEEE Transactions on Automatic Control, vol.12, no.5, pp.585-588, 1967.</ref>

<ref name="GugAB08"> S. Gugercin, A.C. Antoulas, C. Beattie "[http://epubs.siam.org/doi/abs/10.1137/060666123 H2 Model Reduction for Large-Scale Linear Dynamical Systems]", SIAM. J. Matrix Anal. & Appl., vol.30, no.2, pp.609-638, 2008.</ref>

<ref name="FlaBG12"> G. Flagg, C. Beattie, S. Gugercin "[http://www.sciencedirect.com/science/article/pii/S0167691112000576 Convergence of the Iterative Rational Krylov Algorithm]", Systems & Control Letters, vol.61, no.6, pp.688-691, 2012.</ref>

</references>

Projection based MOR

2023-05-11T12:20:53Z

Lund: remove link to list of abbreviations

[[Category:method]]
[[Category:time invariant]]
Consider the linear time invariant system

:<math>
E \frac{dx(t)}{dt}=A x(t)+B u(t), \quad
y(t)=Cx(t), \quad \quad (1)
</math>

as an example.
All the existing model order reduction (MOR) methods are based on projection<ref>Antoulas, A. C. "[http://dx.doi.org/10.1137/1.9780898718713 Approximation of large-scale dynamical systems]". Vol. 6. Society for Industrial and Applied Mathematics, pp.376--377, 2009; ISBN 978-0-89871-529-3</ref>.
That is to find a subspace <math>S_1</math> which approximates the manifold where the state vector <math>x(t)</math> resides.
Afterwards, <math>x(t)</math> is approximated by its projection <math>\hat x(t)</math> in <math>S_1</math>.
The reduced model is produced by [[wikipedia:Petrov–Galerkin_method|Petrov-Galerkin projection]] using a test subspace <math>S_2</math>, or by [[wikipedia:Galerkin_method|Galerkin projection]] using <math>S_1</math> as the test subspace.
Assuming that an orthonormal basis <math>V=(v_1,v_2, \ldots, v_q)</math> of the subspace <math>S_1</math> has been found, then the approximation <math>\hat x(t)</math> in <math>S_1</math> can be represented by the basis as <math>\hat{x} (t)=V z(t)</math>.
Therefore <math>x(t)</math> can be approximated by <math> x(t) \approx V z(t)</math>.
Here <math>z</math> is a vector of length <math>q \ll n</math>.
Once <math>z(t)</math> is computed, an approximate solution <math>\hat x(t)=V z(t)</math> for <math>x(t)</math> can be obtained.
The vector <math>z(t)</math> can be computed from the reduced model, derived by the following two steps.

Step 1. By replacing <math>x</math> in (1) with <math>Vz</math>, we get

:<math>E \frac{d{Vz}}{dt}=A Vz+Bu(t)+e,\quad \hat y(t)=CV z.</math>

Step 2. The residual is denoted as <math>e=AVz+Bu(t)-E \frac{d{Vz}}{dt}</math>. Forcing <math>e=0</math> in a properly chosen subspace <math>S_2</math> of <math>\mathbb {R}^n</math> leads to the Petrov-Galerkin projection: <math>W^T e=0</math>, where the columns of <math>W</math> are the basis of <math>S_2</math>.
Then we have,

:<math>W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t), \quad \hat{y}(t)=CVz.</math>

By defining <math>\hat{E}=W^TEV</math>, <math>\hat {A}=W^TAV, \hat{B}=W^TB</math>, <math>\hat{C}=CV</math>, we get the final reduced model

:<math>\hat{E} \frac{dz(t)}{dt}=\hat{A}z(t)+\hat{B}u(t), \quad
\hat{y}(t)=\hat{C}z(t). \quad \quad (2)
</math>

Notice that the approximation <math>\hat x(t)=Vz(t)</math> of <math>x(t)</math> can be obtained from <math> z(t)</math> by solving the system in (2).
The system in (2) is much smaller than the system in (1) in the sense that there are many less equations in (2) than in (1).
Therefore, the system in (2) is much easier to be solved, which is the so-called reduced model.
In order to save the simulation time of solving (1), the reduced model can be used to replace (1) to attain a fast simulation. Furthermore, the error between the two systems should be within acceptable tolerance.
The error can be measured through the error between the the state vectors <math>x(t), \hat x(t)</math>, or between the output responses <math>y(t), \hat y(t)</math>, or between the transfer functions of the two systems.
It can be seen that once the two matrices <math>W</math> and <math>V</math> have been computed, the reduced model is obtained.
While the Gramian based MOR methods (e.g. [[Balanced Truncation]]) usually compute <math>W</math> different from <math>V</math>, some methods use <math>W=V</math>, e.g. some of the [[Moment-matching method]]s, the [[Reduced Basis PMOR method]]s, and some of the [[POD method]]s etc.. When <math>W=V</math>, Petrov-Galerkin projection becomes Galerkin projection.
MOR methods differ in the computation of the two matrices <math>W</math> and <math>V</math>.
The Gramian based MOR methods compute <math>W</math> and <math>V</math> by the controllability and observability Gramians.
Reduced basis methods and [[List_of_abbreviations#POD|POD]] methods compute <math>V</math> from the snapshots of the state vector <math>x(t)</math> at different time steps (also at the samples of the parameters if the system is parametric). The methods based on moment-matching compute :<math>W</math> and <math>V</math> from the moments of the transfer function.
In eigenvalue based MOR methods, e.g. [[Modal truncation]], the columns of :<math>W</math> and <math>V</math> are eigenvectors or invariant subspaces corresponding to selected eigenvalues of the matrix pair <math>(A,E)</math>.

One common goal of all MOR methods is that the behavior of the reduced model should be sufficiently "close" to that of the original model guaranteed through the above mentioned error measurements.

==References==

<references/>

Branchline Coupler

2023-05-11T12:20:27Z

Lund: remove link to abbreviations

{{preliminary}} 

[[Category:benchmark]]
[[Category:Parametric]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:second differential order]]

==Description==

A '''branchline coupler''' (see Fig. 1) is a microwave semiconductor device, which is simulated by the [http://www.maxwells-equations.com/forms.php#harmonic time-harmonic Maxwell's equation].
A 2-section '''branchline coupler''' consists of four strip line ports, coupled to each other by two transversal bridges.
The energy excited at one port is coupled almost in equal shares to the two opposite ports, when considered as a MIMO-system.
Here, only the [[List_of_abbreviations#SISO|SISO]] case is considered.
The '''branchline coupler''' with <math>0.05mm</math> thickness is placed on a substrate with <math>0.749mm</math> thickness and relative permittivity
<math> \epsilon_r = 2.2 </math> and zero-conductivity <math> \sigma = 0 S/m </math>.
The simulation domain is confined to a <math> 23.6 \times 22 \times 7 mm^3 </math> box.
The metallic ground plane of the device is represented by the electric boundary condition. The magnetic boundary
condition is considered for the other sides of the structures. The discrete input port with source impedance <math>50 \Omega</math>
imposes <math>1 A</math> current as the input. The voltage along the coupled port at the end of the other side of the coupler is
read as the output.

<figure id="fig:branch">
[[File:BranchlineCoupler.png|frame|<caption>Branchline Coupler Model<ref name="hess13"/></caption>]]
</figure>

Considered parameters are the frequency <math>\omega </math> and the relative permeability <math> \mu_r </math> .

The affine form <math> a(u, v; \omega, \mu_r) = \sum_{q=1}^Q \Theta^q(\omega, \mu_r) a^q(u, v) </math> can be established using <math> Q = 2 </math> affine terms.

The discretized bilinear form is <math> a(u, v; \omega, \mu_r) = \sum_{q=1}^Q \Theta^q(\omega, \mu_r) A^q </math>, with matrices <math> A^q </math>.

The matrices corresponding to the bilinear forms <math> a^q( \cdot , \cdot ) </math> as well as the input and output forms and the H(curl) inner product matrix have been assembled
using the [[wikipedia:Finite_Element_Method|Finite Element Method]], resulting in <math>27679</math> degrees of freedom, after removal of boundary conditions.

The coefficient functions are given by:

:<math> \Theta^1(\omega, \mu_r) = \frac{1}{\mu_r} </math>

:<math> \Theta^2(\omega, \mu_r) = -\omega^2. </math>

The parameter domain of interest is <math> \omega \in [1.0, 10.0] * 10^9 Hz</math>, where the factor of <math> 10^9 </math> has already been taken into account
while assembling the matrices, while the material variation occurs between <math> \mu_r \in [0.5, 2.0] </math>. The input functional also has a factor of <math> \omega </math>.

==Data==

The files are numbered according to their appearance in the summation and can be found here:
Part1
[[Media:branchline_part1.zip|branchline_part1.zip]]
Part2
[[Media:branchline_part2.zip|branchline_part2.zip]]
Part3
[[Media:branchline_part3.zip|branchline_part3.zip]]

==Origin==

The models have been developed within the [http://www.moresim4nano.org MoreSim4Nano project].

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_branchcouple,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Branchline Coupler},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Branchline_Coupler}</nowiki>,
year = {2013}
}

* For the background on the benchmark:
@ARTICLE{morHesB13,
author = {M.~W. Hess and P. Benner},
title = {Fast Evaluation of Time-Harmonic {M}axwell's
Equations Using the Reduced Basis Method},
journal = {{IEEE} Trans. Microw. Theory Techn.},
year = 2013,
volume = 61,
number = 6,
pages = {2265--2274},
doi = {10.1109/TMTT.2013.2258167}
}

==References==

<references>

<ref name="hess13">M. W. Hess, P. Benner, "[https://doi.org/10.1109/TMTT.2013.2258167 Fast Evaluation of Time-Harmonic Maxwell's Equations Using the Reduced Basis Method]", IEEE Transactions on Microwave Theory and Techniques, 61(6): 2265--2274, 2013.</ref>

</references>

==Contact==

'' [[User:hessm|Martin Hess]]''

Glossary

2023-05-11T12:15:44Z

Lund: Updated from list of abbreviations

{{preliminary}} 

This page contains a glossary of terms related to the [[MORB]] collection, benchmark.csv, the [[Search Tool]], and this wiki as a whole.

Note that all boolean fields are really quadruple valued; in addition to 0 (false) and 1 (true), other options include the following:
* an empty field, indicating a lack of knowledge, e.g., either no attempt has been made to figure out the value, or none of the available methods have been successful in determining the value thus far.
* NaN, indicating either that the property does not apply or is otherwise impossible to determine

{| class="wikitable" style="background-color:#FFF;"
|- style="font-weight:bold; vertical-align:middle;"
! Abbreviation / Attribute / Term
! Description
|- style="vertical-align:middle;"
| <code>_comments</code>
| (char) internal comments
|- style="vertical-align:middle;"
| <code>_dataSource</code>
| (char) where the data was pulled from during processing: [https://gitlab.mpi-magdeburg.mpg.de/himpe/unimat unimat], [https://sites.google.com/site/rommes/software rommes], or [https://modelreduction.org morwiki]
|- style="vertical-align:middle;"
| <code>_sourceFilename</code>
| (char) filename of data used during processing
|- style="vertical-align:middle;"
| AP
| affine parametric; parameters of any component are linear combinations of fixed matrices
|- style="vertical-align:middle;"
| <code>components</code>
| (char) matrices stored in .mat file; see also [[Models]]
|- style="vertical-align:middle;"
| <code>cond*</code>
| (list of float) the numerically computed ratio between the largest and smallest singular values of the matrix * (including 0); Inf is treated as a float in MATLAB; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>creator</code>
| (list of char) ORCID(s) of original author(s) or creator(s) of dataset
|- style="vertical-align:middle;"
| DAE
| [https://en.wikipedia.org/wiki/Differential_algebraic_equation differential algebraic equation]
|- style="vertical-align:middle;"
| <code>daeDiffIndex</code>
| (int) DAE differentiation index; roughly the number of differentiations required to make a system an ODE
|- style="vertical-align:middle;"
| DEIM
| discrete empirical interpolation method
|- style="vertical-align:middle;"
| <code>directory</code>
| (char) folder / collection to which a benchmark belongs
|- style="vertical-align:middle;"
| DoF
| [https://en.wikipedia.org/wiki/Degrees_of_Freedom degrees of freedom]
|- style="vertical-align:middle;"
| DSPMR
| [[Dominant_Subspaces|dominant subspace projection model reduction]]
|- style="vertical-align:middle;"
| <code>editor</code>
| (list of char) ORCID(s) of editor(s) who converted data to new .mat format
|- style="vertical-align:middle;"
| FEM
| [https://en.wikipedia.org/wiki/Finite_Element_Method finite element method]
|- style="vertical-align:middle;"
| <code>filename</code>
| (char) filename and benchmark ID
|- style="vertical-align:middle;"
| FOM
| full order model
|- style="vertical-align:middle;"
| FOS
| first-order system
|- style="vertical-align:middle;"
| <code>is*CholAble</code>
| (list of bool) the matrix * is symmetric positive definite, i.e., it admits a Cholesky factorization (and is therefore "Cholesky-able"); note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>is*Sparse</code>
| (list of bool) the matrix * is stored in a sparse format; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>is*Symm</code>
| (list of bool) the matrix is symmetric; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>isContractive</code>
| (bool) a system that is dissipative w.r.t. <math>s(u(t),y(t)) = ||u(t)||^2 - ||y(t)||^2</math>
|- style="vertical-align:middle;"
| <code>isDAE</code>
| (bool) differential algebraic equation; an LTI DAE has a nonzero, singular E component
|- style="vertical-align:middle;"
| <code>isPassive</code>
| (bool) a system that is dissipative w.r.t s(u(t),y(t)) = u(t)^T y(t)
|- style="vertical-align:middle;"
| <code>isSquare</code>
| (bool) nInputs = nOutputs
|- style="vertical-align:middle;"
| <code>isStable</code>
| (int) numerically computed; 0 = no stability 1 = asymptotically stable; i.e., all finite eigenvalues of the pencil <math>sE-A</math> (<math>E = I</math>, if not otherwise specified) have negative real part; consequently, the solution of the corresponding LTI with <math>u(t) = 0</math> tends to 0 as <math>t \to \infty</math> 2 = ? 3 = ? (other stability definitions added and defined as necessary)
|- style="vertical-align:middle;"
| <code>isStateSpaceSymm</code>
| (bool) state-space symmetric: <math>A = A^T</math>, <math>C = B^T</math>, <math>E = E^T</math>, <math>D = D^T</math>; trivially implies <code>isSysSymm</code>; similarly defined for SOS, but require instead that <math>M = M^T</math>, <math>C_v = 0, C_p = B^T</math>, and <math>K = K^T</math>
|- style="vertical-align:middle;"
| <code>isSysSymm</code>
| (bool) <math>H(s) = C (sE -A)^{-1} B+D</math> is symmetric; equivalently, the system's Markov parameters are symmetric For a distinction between symmetry and state-space symmetry, see Def 2.1 in [https://www.sciencedirect.com/science/article/pii/S0167691198000243 this paper]. [https://doi.org/10.1109/PROC.1983.12688 Another useful source]
|- style="vertical-align:middle;"
| <code>license</code>
| (char) license associated to data
|- style="vertical-align:middle;"
| LTI
| linear time-invariant
|- style="vertical-align:middle;"
| LTV
| linear time-varying
|- style="vertical-align:middle;"
| MIMO
| [https://en.wikipedia.org/wiki/MIMO multiple-input, multiple-output]
|- style="vertical-align:middle;"
| MOR
| [https://modelreduction.org/ model order reduction]
|- style="vertical-align:middle;"
| <code>MORWikiLink</code>
| (char) link to MORWiki page
|- style="vertical-align:middle;"
| <code>MORWikiPageName</code>
| (char) fixed page name with plain text description of system in the MORWiki
|- style="vertical-align:middle;"
| <code>nInputs</code>
| (int) number of inputs
|- style="vertical-align:middle;"
| NL
| nonlinear; a nonlinear system depends nonlinearly on <math>x(t)</math>
|- style="vertical-align:middle;"
| <code>nnz*</code>
| (list of int) when the matrix * is sparse, the number of nonzeros; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>nOutputs</code>
| (int) number of outputs
|- style="vertical-align:middle;"
| <code>nParameters</code>
| (int) maximum number of parameters in a parametric system
|- style="vertical-align:middle;"
| <code>nStates</code>
| (int) number of states
|- style="vertical-align:middle;"
| <code>nUnstabPoles</code>
| (int) the number of poles with nonnegative real part, computed numerically
|- style="vertical-align:middle;"
| ODE
| [https://en.wikipedia.org/wiki/Ordinary_differential_equation ordinary differential equation]
|- style="vertical-align:middle;"
| PDE
| [https://en.wikipedia.org/wiki/Partial_differential_equation partial differential equation]
|- style="vertical-align:middle;"
| PMOR
| parametric model order reduction
|- style="vertical-align:middle;"
| POD
| proper orthogonal decomposition
|- style="vertical-align:middle;"
| QBS
| quadratic bilinear system
|- style="vertical-align:middle;"
| RBM
| [https://pnp.mathematik.uni-stuttgart.de/ians/haasdonk/publications/RBtutorial_preprint_update_with_header.pdf reduced basis method]
|- style="vertical-align:middle;"
| ROM
| reduced order model
|- style="vertical-align:middle;"
| SISO
| [https://en.wikipedia.org/wiki/Single-Input_and_Single-Output single-input, single-output]
|- style="vertical-align:middle;"
| SOS
| second-order system
|- style="vertical-align:middle;"
| SVD
| [https://en.wikipedia.org/wiki/Singular_Value_Decomposition singular value decomposition]
|- style="vertical-align:middle;"
| <code>systemClass</code>
| (char) model realization; see also [[Models]]
|- style="vertical-align:middle;"
| <code>zenodoDOI</code>
| (char) zenodo DOI (not the full URL!)
|}

Modified Gyroscope

2023-05-11T08:45:29Z

Lund: /* Data */ Clarify how matrix names are changed for MORB

[[Category:benchmark]]
[[Category:ODE]]
[[Category:linear]]
[[Category:Parametric]]
[[Category:affine parameter representation]]
[[Category:time invariant]]
[[Category:second differential order]]

==Description==
<figure id="fig:gyro">[[File:Gyroscope.jpg|300px|thumb|right|<caption>Schematic representation of the gyroscope.</caption>]]</figure>

The device is a [[wikipedia:Vibrating_structure_gyroscope#MEMS_gyroscopes|MEMS gyroscope]] based on the butterfly gyroscope <ref name="lienemann2004"/> developed at the [http://www.imego.com/ Imego institute] in Gothenburg,
Sweden (see also: [[Butterfly_Gyroscope|Butterfly Gyroscope]], where a non-parametrized model for the device is given).
A [[wikipedia:Gyroscope|gyroscope]] is a device used to measure angular rates in up to three axes.

The basic working principle of the '''gyroscope''' can be described as follows, see also <ref name="Moo07"/>.
Without applied external rotation, the paddles vibrate in phase with the function <math>z(t),</math> see Fig. 1.
Under the influence of an external rotation about the <math>x</math>-axis (drawn in red),
an additional force due to the Coriolis acceleration acts upon the paddles.
This force leads to an additional small out-of-phase vibration between two paddles on the same side of the bearing.
This out-of phase vibration is measured as the difference of the <math>z</math>-displacement of the nodes with the red dots.
Thus, measuring the displacement of two adjacent paddles, the rotation velocity can be ascertained.

==Motivation==

When planning for and making decisions on future improvements of the butterfly gyroscope, it is of importance to improve the efficiency of the gyro simulations. Repeated analysis 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 run time for repeated simulations.

==The Parametrized Model==

Two parameters are of special interest for the model.
The first one is the quantity that is to be sensed, the rotation velocity <math>\theta</math> around the <math>x</math>-axes.
The second parameter is the width of the bearing, <math>d</math>.
The parametrized system below is obtained by finite element discretization of the parametrized model (in the form of partial differential equations) for the '''gyroscope'''.
The details of constructing the parametrized system can be found in <ref name="Moo07"/>.
The system is of the following form:

:<math>
\begin{align}
M(d)\ddot{x}(t) +D(d,\theta)\dot{x}(t) +T(d)x(t) &= B, \\
y(t) &=Cx(t).
\end{align}
</math>

Here,

* <math>M(d)=M_1 + dM_2\in \mathbb R^{n\times n}</math> is the mass matrix,
* <math>D(d,\theta)=\theta(D_1 + d D_2)\in \mathbb R^{n\times n}</math> is the damping matrix,
* <math>T(d)=T_1+(1/d)T_2+dT_3\in \mathbb R^{n\times n}</math> is the stiffness matrix,
* <math>B \in \mathbb R^{n \times 1}</math> is the load vector,
* <math>C \in \mathbb R^{1 \times n}</math> is the output matrix,
* <math>x \in \mathbb R^{n}</math> is the state vector,
* and <math>y \in \mathbb R </math> is the output response.

The quantity of interest <math>y</math> of the system is <math>\delta z(t)</math>, which is the difference of the displacement <math>z(t)</math> between the two red markings on the ''east'' side of the bearing (see <xr id="fig:gyro"/>).

The parameters of the system, <math>d</math> and <math>\theta</math>, represent the width of the bearing(<math>d</math>) and the rotation velocity along the <math>x</math>-axis (<math>\theta</math>), with the ranges: <math>\theta\in [10^{-7}, 10^{-5}]</math> and <math>d\in [1,2]</math>.

The device works in the frequency range <math>f \in [0.025, 0.25]</math>MHz and the degrees of freedom are <math>n = 17913</math>.

==Data==

The model is generated in ANSYS.
The system matrices <math>M_1, \, M_2, \, D_1, \, D_2, \, T_1, \, T_2, \, T_3</math>, and <math> B</math> are in the [http://math.nist.gov/MatrixMarket/ MatrixMarket format], and can be downloaded here: [[Media: Gyroscope_modi.tgz|Gyroscope_modi.tgz]]. The matrix <math>C</math> defines the output, which has zeros on all the entries, except on the 2315th entry, where the value is <math>-1</math>, and on the 5806th entry, the value is <math>1</math>;
in MATLAB notation, it is <tt>C(1,2315) = -1</tt> and <tt>C(1,5806) = 1</tt>.

The matrices have been reindexed for [[MORB]], as shown in the next section. In particular, <math>D_i</math> is denoted instead by <math>E_i</math>, and <math>T_i</math> by <math>K_i</math>.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
(M_1 + d M_2)\ddot{x}(t) + \theta(E_1 + d E_2) \dot{x}(t) + (K_1 + d^{-1} K_2 + d K_3)x(t) &=& B \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>M_{1,2}, E_{1,2}, K_{1,2,3} \in \mathbb{R}^{17931 \times 17931}</math>,
<math>B \in \mathbb{R}^{17931 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 17931}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_modgyro,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Modified Gyroscope},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Modified_Gyroscope}</nowiki>,
year = 2018
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morMoo07 morMoo07] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morMoo07 BibTeX])

==References==

<references>

<ref name="lienemann2004">J. Lienemann, D. Billger, E. B. Rudnyi, A. Greiner, J. G. Korvink, "[http://www.nsti.org/procs/Nanotech2004v2/6/W58.01 MEMS Compact Modeling Meets Model Order Reduction: Examples of the Application of Arnoldi Methods to Microsystem Devices]", TechConnect Briefs (Technical Proceedings of the 2004 NSTI Nanotechnology Conference and Trade Show, Volume 2): 303--306, 2004.</ref>

<ref name="Moo07">C. Moosmann, "[http://www.freidok.uni-freiburg.de/volltexte/3971/ ParaMOR Model Order Reduction for parameterized MEMS applications]", PhD thesis, Department of Microsystems Engineering, University of Freiburg, Freiburg, Germany 2007.</ref>

</references>

==Contact==

'' [[User:Feng]] ''

Modified Gyroscope

2023-05-11T08:43:49Z

Lund: /* The Parametrized Model */ Update matrices to match what is stored in matrix market file

[[Category:benchmark]]
[[Category:ODE]]
[[Category:linear]]
[[Category:Parametric]]
[[Category:affine parameter representation]]
[[Category:time invariant]]
[[Category:second differential order]]

==Description==
<figure id="fig:gyro">[[File:Gyroscope.jpg|300px|thumb|right|<caption>Schematic representation of the gyroscope.</caption>]]</figure>

The device is a [[wikipedia:Vibrating_structure_gyroscope#MEMS_gyroscopes|MEMS gyroscope]] based on the butterfly gyroscope <ref name="lienemann2004"/> developed at the [http://www.imego.com/ Imego institute] in Gothenburg,
Sweden (see also: [[Butterfly_Gyroscope|Butterfly Gyroscope]], where a non-parametrized model for the device is given).
A [[wikipedia:Gyroscope|gyroscope]] is a device used to measure angular rates in up to three axes.

The basic working principle of the '''gyroscope''' can be described as follows, see also <ref name="Moo07"/>.
Without applied external rotation, the paddles vibrate in phase with the function <math>z(t),</math> see Fig. 1.
Under the influence of an external rotation about the <math>x</math>-axis (drawn in red),
an additional force due to the Coriolis acceleration acts upon the paddles.
This force leads to an additional small out-of-phase vibration between two paddles on the same side of the bearing.
This out-of phase vibration is measured as the difference of the <math>z</math>-displacement of the nodes with the red dots.
Thus, measuring the displacement of two adjacent paddles, the rotation velocity can be ascertained.

==Motivation==

When planning for and making decisions on future improvements of the butterfly gyroscope, it is of importance to improve the efficiency of the gyro simulations. Repeated analysis 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 run time for repeated simulations.

==The Parametrized Model==

Two parameters are of special interest for the model.
The first one is the quantity that is to be sensed, the rotation velocity <math>\theta</math> around the <math>x</math>-axes.
The second parameter is the width of the bearing, <math>d</math>.
The parametrized system below is obtained by finite element discretization of the parametrized model (in the form of partial differential equations) for the '''gyroscope'''.
The details of constructing the parametrized system can be found in <ref name="Moo07"/>.
The system is of the following form:

:<math>
\begin{align}
M(d)\ddot{x}(t) +D(d,\theta)\dot{x}(t) +T(d)x(t) &= B, \\
y(t) &=Cx(t).
\end{align}
</math>

Here,

* <math>M(d)=M_1 + dM_2\in \mathbb R^{n\times n}</math> is the mass matrix,
* <math>D(d,\theta)=\theta(D_1 + d D_2)\in \mathbb R^{n\times n}</math> is the damping matrix,
* <math>T(d)=T_1+(1/d)T_2+dT_3\in \mathbb R^{n\times n}</math> is the stiffness matrix,
* <math>B \in \mathbb R^{n \times 1}</math> is the load vector,
* <math>C \in \mathbb R^{1 \times n}</math> is the output matrix,
* <math>x \in \mathbb R^{n}</math> is the state vector,
* and <math>y \in \mathbb R </math> is the output response.

The quantity of interest <math>y</math> of the system is <math>\delta z(t)</math>, which is the difference of the displacement <math>z(t)</math> between the two red markings on the ''east'' side of the bearing (see <xr id="fig:gyro"/>).

The parameters of the system, <math>d</math> and <math>\theta</math>, represent the width of the bearing(<math>d</math>) and the rotation velocity along the <math>x</math>-axis (<math>\theta</math>), with the ranges: <math>\theta\in [10^{-7}, 10^{-5}]</math> and <math>d\in [1,2]</math>.

The device works in the frequency range <math>f \in [0.025, 0.25]</math>MHz and the degrees of freedom are <math>n = 17913</math>.

==Data==

The model is generated in ANSYS.
The system matrices <math>M, \, M_1, \, E_1, \, E_2, \, K, \, K_1, \,K_2</math>, and <math> B</math> are in the [http://math.nist.gov/MatrixMarket/ MatrixMarket format], and can be downloaded here: [[Media: Gyroscope_modi.tgz|Gyroscope_modi.tgz]].
The matrix <math>C</math> defines the output, which has zeros on all the entries, except on the 2315th entry, where the value is <math>-1</math>, and on the 5806th entry, the value is <math>1</math>;
in MATLAB notation, it is <tt>C(1,2315) = -1</tt> and <tt>C(1,5806) = 1</tt>.

The matrices have been reindexed for [[MORB]], as shown in the next section.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
(M_1 + d M_2)\ddot{x}(t) + \theta(E_1 + d E_2) \dot{x}(t) + (K_1 + d^{-1} K_2 + d K_3)x(t) &=& B \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>M_{1,2}, E_{1,2}, K_{1,2,3} \in \mathbb{R}^{17931 \times 17931}</math>,
<math>B \in \mathbb{R}^{17931 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 17931}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_modgyro,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Modified Gyroscope},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Modified_Gyroscope}</nowiki>,
year = 2018
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morMoo07 morMoo07] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morMoo07 BibTeX])

==References==

<references>

<ref name="lienemann2004">J. Lienemann, D. Billger, E. B. Rudnyi, A. Greiner, J. G. Korvink, "[http://www.nsti.org/procs/Nanotech2004v2/6/W58.01 MEMS Compact Modeling Meets Model Order Reduction: Examples of the Application of Arnoldi Methods to Microsystem Devices]", TechConnect Briefs (Technical Proceedings of the 2004 NSTI Nanotechnology Conference and Trade Show, Volume 2): 303--306, 2004.</ref>

<ref name="Moo07">C. Moosmann, "[http://www.freidok.uni-freiburg.de/volltexte/3971/ ParaMOR Model Order Reduction for parameterized MEMS applications]", PhD thesis, Department of Microsystems Engineering, University of Freiburg, Freiburg, Germany 2007.</ref>

</references>

==Contact==

'' [[User:Feng]] ''

Glossary

2023-05-10T11:20:04Z

Lund: add state-space symm definition for SOS

{{preliminary}} 

This page contains a glossary of terms related to the [[MORB]] collection, benchmark.csv, the [[Search Tool]], and this wiki as a whole.

Note that all boolean fields are really quadruple valued; in addition to 0 (false) and 1 (true), other options include the following:
* an empty field, indicating a lack of knowledge, e.g., either no attempt has been made to figure out the value, or none of the available methods have been successful in determining the value thus far.
* NaN, indicating either that the property does not apply or is otherwise impossible to determine

{| class="wikitable" style="background-color:#FFF;"
|- style="font-weight:bold; vertical-align:middle;"
! Abbreviation / Attribute / Term
! Description
|- style="vertical-align:middle;"
| <code>_comments</code>
| (char) internal comments
|- style="vertical-align:middle;"
| <code>_dataSource</code>
| (char) where the data was pulled from during processing: [[https://gitlab.mpi-magdeburg.mpg.de/himpe/unimat unimat]], [[https://sites.google.com/site/rommes/software rommes]], or [[https://modelreduction.org morwiki]]
|- style="vertical-align:middle;"
| <code>_sourceFilename</code>
| (char) filename of data used during processing
|- style="vertical-align:middle;"
| AP
| affine parametric; parameters of any component are linear combinations of fixed matrices
|- style="vertical-align:middle;"
| <code>components</code>
| (char) matrices stored in .mat file; see also [[Models]]
|- style="vertical-align:middle;"
| <code>cond*</code>
| (list of float) the numerically computed ratio between the largest and smallest singular values of the matrix * (including 0); Inf is treated as a float in MATLAB; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>creator</code>
| (list of char) ORCID(s) of original author(s) or creator(s) of dataset
|- style="vertical-align:middle;"
| DAE
| differential algebraic equation; subset of LTI
|- style="vertical-align:middle;"
| <code>daeDiffIndex</code>
| (int) DAE differentiation index; roughly the number of differentiations required to make a system an ODE
|- style="vertical-align:middle;"
| DEIM
| discrete empirical interpolation method
|- style="vertical-align:middle;"
| <code>directory</code>
| (char) folder / collection to which a benchmark belongs
|- style="vertical-align:middle;"
| DSPMR
| dominant subspace projection model reduction
|- style="vertical-align:middle;"
| <code>editor</code>
| (list of char) ORCID(s) of editor(s) who converted data to new .mat format
|- style="vertical-align:middle;"
| <code>filename</code>
| (char) filename and benchmark ID
|- style="vertical-align:middle;"
| FOM
| full order model
|- style="vertical-align:middle;"
| FOS
| first-order system
|- style="vertical-align:middle;"
| <code>is*CholAble</code>
| (list of bool) the matrix * is symmetric positive definite, i.e., it admits a Cholesky factorization (and is therefore "Cholesky-able"); note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>is*Sparse</code>
| (list of bool) the matrix * is stored in a sparse format; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>is*Symm</code>
| (list of bool) the matrix is symmetric; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>isContractive</code>
| (bool) a system that is dissipative w.r.t. <math>s(u(t),y(t)) = ||u(t)||^2 - ||y(t)||^2</math>
|- style="vertical-align:middle;"
| <code>isDAE</code>
| (bool) differential algebraic equation; an LTI DAE has a nonzero, singular E component
|- style="vertical-align:middle;"
| <code>isPassive</code>
| (bool) a system that is dissipative w.r.t s(u(t),y(t)) = u(t)^T y(t)
|- style="vertical-align:middle;"
| <code>isSquare</code>
| (bool) nInputs = nOutputs
|- style="vertical-align:middle;"
| <code>isStable</code>
| (int) numerically computed; 0 = no stability 1 = asymptotically stable; i.e., all finite eigenvalues of the pencil <math>sE-A</math> (<math>E = I</math>, if not otherwise specified) have negative real part; consequently, the solution of the corresponding LTI with <math>u(t) = 0</math> tends to 0 as <math>t \to \infty</math> 2 = ? 3 = ? (other stability definitions added and defined as necessary)
|- style="vertical-align:middle;"
| <code>isStateSpaceSymm</code>
| (bool) state-space symmetric: <math>A = A^T</math>, <math>C = B^T</math>, <math>E = E^T</math>, <math>D = D^T</math>; trivially implies <code>isSysSymm</code>; similarly defined for SOS, but require instead that <math>M = M^T</math>, <math>C_v = 0, C_p = B^T</math>, and <math>K = K^T</math>
|- style="vertical-align:middle;"
| <code>isSysSymm</code>
| (bool) <math>H(s) = C (sE -A)^{-1} B+D</math> is symmetric; equivalently, the system's Markov parameters are symmetric For a distinction between symmetry and state-space symmetry, see Def 2.1 in [[https://www.sciencedirect.com/science/article/pii/S0167691198000243 this paper]]. [[https://doi.org/10.1109/PROC.1983.12688 Another useful source]]
|- style="vertical-align:middle;"
| <code>license</code>
| (char) license associated to data
|- style="vertical-align:middle;"
| LTI
| linear time-invariant
|- style="vertical-align:middle;"
| LTV
| linear time-varying
|- style="vertical-align:middle;"
| MIMO
| multiple-input, multiple-output
|- style="vertical-align:middle;"
| MOR
| model order reduction
|- style="vertical-align:middle;"
| <code>MORWikiLink</code>
| (char) link to MORWiki page
|- style="vertical-align:middle;"
| <code>MORWikiPageName</code>
| (char) fixed page name with plain text description of system in the MORWiki
|- style="vertical-align:middle;"
| <code>nInputs</code>
| (int) number of inputs
|- style="vertical-align:middle;"
| NL
| nonlinear; a nonlinear system depends nonlinearly on <math>x(t)</math>
|- style="vertical-align:middle;"
| <code>nnz*</code>
| (list of int) when the matrix * is sparse, the number of nonzeros; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>nOutputs</code>
| (int) number of outputs
|- style="vertical-align:middle;"
| <code>nParameters</code>
| (int) maximum number of parameters in a parametric system
|- style="vertical-align:middle;"
| <code>nStates</code>
| (int) number of states
|- style="vertical-align:middle;"
| <code>nUnstabPoles</code>
| (int) the number of poles with nonnegative real part, computed numerically
|- style="vertical-align:middle;"
| ODE
| ordinary differential equation
|- style="vertical-align:middle;"
| PDE
| partial differential equation
|- style="vertical-align:middle;"
| PMOR
| parametric model order reduction
|- style="vertical-align:middle;"
| POD
| proper orthogonal decomposition
|- style="vertical-align:middle;"
| QBS
| quadratic bilinear system
|- style="vertical-align:middle;"
| RBM
| reduced basis model
|- style="vertical-align:middle;"
| ROM
| reduced order model
|- style="vertical-align:middle;"
| SISO
| single-input, single-output
|- style="vertical-align:middle;"
| SOS
| second-order system
|- style="vertical-align:middle;"
| <code>systemClass</code>
| (char) model realization; see also [[Models]]
|- style="vertical-align:middle;"
| <code>zenodoDOI</code>
| (char) zenodo DOI (not the full URL!)
|}

Glossary

2023-05-10T11:13:27Z

Lund: typo

{{preliminary}} 

This page contains a glossary of terms related to the [[MORB]] collection, benchmark.csv, the [[Search Tool]], and this wiki as a whole.

Note that all boolean fields are really quadruple valued; in addition to 0 (false) and 1 (true), other options include the following:
* an empty field, indicating a lack of knowledge, e.g., either no attempt has been made to figure out the value, or none of the available methods have been successful in determining the value thus far.
* NaN, indicating either that the property does not apply or is otherwise impossible to determine

{| class="wikitable" style="background-color:#FFF;"
|- style="font-weight:bold; vertical-align:middle;"
! Abbreviation / Attribute / Term
! Description
|- style="vertical-align:middle;"
| <code>_comments</code>
| (char) internal comments
|- style="vertical-align:middle;"
| <code>_dataSource</code>
| (char) where the data was pulled from during processing: [[https://gitlab.mpi-magdeburg.mpg.de/himpe/unimat unimat]], [[https://sites.google.com/site/rommes/software rommes]], or [[https://modelreduction.org morwiki]]
|- style="vertical-align:middle;"
| <code>_sourceFilename</code>
| (char) filename of data used during processing
|- style="vertical-align:middle;"
| AP
| affine parametric; parameters of any component are linear combinations of fixed matrices
|- style="vertical-align:middle;"
| <code>components</code>
| (char) matrices stored in .mat file; see also [[Models]]
|- style="vertical-align:middle;"
| <code>cond*</code>
| (list of float) the numerically computed ratio between the largest and smallest singular values of the matrix * (including 0); Inf is treated as a float in MATLAB; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>creator</code>
| (list of char) ORCID(s) of original author(s) or creator(s) of dataset
|- style="vertical-align:middle;"
| DAE
| differential algebraic equation; subset of LTI
|- style="vertical-align:middle;"
| <code>daeDiffIndex</code>
| (int) DAE differentiation index; roughly the number of differentiations required to make a system an ODE
|- style="vertical-align:middle;"
| DEIM
| discrete empirical interpolation method
|- style="vertical-align:middle;"
| <code>directory</code>
| (char) folder / collection to which a benchmark belongs
|- style="vertical-align:middle;"
| DSPMR
| dominant subspace projection model reduction
|- style="vertical-align:middle;"
| <code>editor</code>
| (list of char) ORCID(s) of editor(s) who converted data to new .mat format
|- style="vertical-align:middle;"
| <code>filename</code>
| (char) filename and benchmark ID
|- style="vertical-align:middle;"
| FOM
| full order model
|- style="vertical-align:middle;"
| FOS
| first-order system
|- style="vertical-align:middle;"
| <code>is*CholAble</code>
| (list of bool) the matrix * is symmetric positive definite, i.e., it admits a Cholesky factorization (and is therefore "Cholesky-able"); note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>is*Sparse</code>
| (list of bool) the matrix * is stored in a sparse format; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>is*Symm</code>
| (list of bool) the matrix is symmetric; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>isContractive</code>
| (bool) a system that is dissipative w.r.t. <math>s(u(t),y(t)) = ||u(t)||^2 - ||y(t)||^2</math>
|- style="vertical-align:middle;"
| <code>isDAE</code>
| (bool) differential algebraic equation; an LTI DAE has a nonzero, singular E component
|- style="vertical-align:middle;"
| <code>isPassive</code>
| (bool) a system that is dissipative w.r.t s(u(t),y(t)) = u(t)^T y(t)
|- style="vertical-align:middle;"
| <code>isSquare</code>
| (bool) nInputs = nOutputs
|- style="vertical-align:middle;"
| <code>isStable</code>
| (int) numerically computed; 0 = no stability 1 = asymptotically stable; i.e., all finite eigenvalues of the pencil <math>sE-A</math> (<math>E = I</math>, if not otherwise specified) have negative real part; consequently, the solution of the corresponding LTI with <math>u(t) = 0</math> tends to 0 as <math>t \to \infty</math> 2 = ? 3 = ? (other stability definitions added and defined as necessary)
|- style="vertical-align:middle;"
| <code>isStateSpaceSymm</code>
| (bool) <math>A = A^T</math>, <math>C = B^T</math>, <math>E = E^T</math>, <math>D = D^T</math>; trivially implies <code>isSysSymm</code>
|- style="vertical-align:middle;"
| <code>isSysSymm</code>
| (bool) <math>H(s) = C (sE -A)^{-1} B+D</math> is symmetric; equivalently, the system's Markov parameters are symmetric For a distinction between symmetry and state-space symmetry, see Def 2.1 in [[https://www.sciencedirect.com/science/article/pii/S0167691198000243 this paper]]. [[https://doi.org/10.1109/PROC.1983.12688 Another useful source]]
|- style="vertical-align:middle;"
| <code>license</code>
| (char) license associated to data
|- style="vertical-align:middle;"
| LTI
| linear time-invariant
|- style="vertical-align:middle;"
| LTV
| linear time-varying
|- style="vertical-align:middle;"
| MIMO
| multiple-input, multiple-output
|- style="vertical-align:middle;"
| MOR
| model order reduction
|- style="vertical-align:middle;"
| <code>MORWikiLink</code>
| (char) link to MORWiki page
|- style="vertical-align:middle;"
| <code>MORWikiPageName</code>
| (char) fixed page name with plain text description of system in the MORWiki
|- style="vertical-align:middle;"
| <code>nInputs</code>
| (int) number of inputs
|- style="vertical-align:middle;"
| NL
| nonlinear; a nonlinear system depends nonlinearly on <math>x(t)</math>
|- style="vertical-align:middle;"
| <code>nnz*</code>
| (list of int) when the matrix * is sparse, the number of nonzeros; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>nOutputs</code>
| (int) number of outputs
|- style="vertical-align:middle;"
| <code>nParameters</code>
| (int) maximum number of parameters in a parametric system
|- style="vertical-align:middle;"
| <code>nStates</code>
| (int) number of states
|- style="vertical-align:middle;"
| <code>nUnstabPoles</code>
| (int) the number of poles with nonnegative real part, computed numerically
|- style="vertical-align:middle;"
| ODE
| ordinary differential equation
|- style="vertical-align:middle;"
| PDE
| partial differential equation
|- style="vertical-align:middle;"
| PMOR
| parametric model order reduction
|- style="vertical-align:middle;"
| POD
| proper orthogonal decomposition
|- style="vertical-align:middle;"
| QBS
| quadratic bilinear system
|- style="vertical-align:middle;"
| RBM
| reduced basis model
|- style="vertical-align:middle;"
| ROM
| reduced order model
|- style="vertical-align:middle;"
| SISO
| single-input, single-output
|- style="vertical-align:middle;"
| SOS
| second-order system
|- style="vertical-align:middle;"
| <code>systemClass</code>
| (char) model realization; see also [[Models]]
|- style="vertical-align:middle;"
| <code>zenodoDOI</code>
| (char) zenodo DOI (not the full URL!)
|}

Glossary

2023-05-09T16:00:00Z

Lund: add description of booleans

{{preliminary}} 

This page contains a glossary of terms related to the [[MORB]] collection, benchmark.csv, the [[Search Tool]], and this wiki as a whole.

Note that all boolean fields are really quadruple valued; in addition to 0 (false) and 1 (true), other options include the following:
* an empty field, indicating a lack of knowledge, e.g., either no attempt has been made to figure out the value, or none of the available methods have been successful in determining the value thus far.
* NaN, indicating either that the property does not apply or is otherwise impossible to determine

{| class="wikitable" style="background-color:#FFF;"
|- style="font-weight:bold; vertical-align:middle;"
! Abbreviation / Attribute / Term
! Description
|- style="vertical-align:middle;"
| <code>_comments</code>
| (char) internal comments
|- style="vertical-align:middle;"
| <code>_dataSource</code>
| (char) where the data was pulled from during processing: [[https://gitlab.mpi-magdeburg.mpg.de/himpe/unimat unimat]], [[https://sites.google.com/site/rommes/software rommes]], or [[https://modelreduction.org morwiki]]
|- style="vertical-align:middle;"
| <code>_sourceFilename</code>
| (char) filename of data used during processing
|- style="vertical-align:middle;"
| AP
| affine parametric; parameters of any component are linear combinations of fixed matrices
|- style="vertical-align:middle;"
| <code>components</code>
| (char) matrices stored in .mat file; see also [[Models]]
|- style="vertical-align:middle;"
| <code>cond*</code>
| (list of float) the numerically computed ratio between the largest and smallest singular values of the matrix * (including 0); Inf is treated as a float in MATLAB; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>creator</code>
| (list of char) ORCID(s) of original author(s) or creator(s) of dataset
|- style="vertical-align:middle;"
| DAE
| differential algebraic equation; subset of LTI
|- style="vertical-align:middle;"
| <code>daeDiffIndex</code>
| (int) DAE differentiation index; roughly the number of differentiations required to make a system an ODE
|- style="vertical-align:middle;"
| DEIM
| discrete empirical interpolation method
|- style="vertical-align:middle;"
| <code>directory</code>
| (char) folder / collection to which a benchmark belongs
|- style="vertical-align:middle;"
| DSPMR
| dominant subspace projection model reduction
|- style="vertical-align:middle;"
| <code>editor</code>
| (list of char) ORCID(s) of editor(s) who converted data to new .mat format
|- style="vertical-align:middle;"
| <code>filename</code>
| (char) filename and benchmark ID
|- style="vertical-align:middle;"
| FOM
| full order model
|- style="vertical-align:middle;"
| FOS
| first-order system
|- style="vertical-align:middle;"
| <code>is*CholAble</code>
| (list of bool) the matrix * is symmetric positive definite, i.e., it admits a Cholesky factorization (and is therefore "Cholesky-able"); note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>is*Sparse</code>
| (list of bool) the matrix * is stored in a sparse format; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>is*Symm</code>
| (list of bool) the matrix is symmetric; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>isContractive</code>
| (bool) a system that is dissipative w.r.t. <math>s(u(t),y(t)) = ||u(t)||^2 - ||y(t)||^2</math>
|- style="vertical-align:middle;"
| <code>isDAE</code>
| (bool) differential algebraic equation; an LTI DAE has a nonzero, singular E component
|- style="vertical-align:middle;"
| <code>isPassive</code>
| (bool) a system that is dissipative w.r.t s(u(t),y(t)) = u(t)^T y(t)
|- style="vertical-align:middle;"
| <code>isSquare</code>
| (bool) nInputs = nOutputs
|- style="vertical-align:middle;"
| <code>isStable</code>
| (int) numerically computed; 0 = no stability 1 = asymptotically stable; i.e., all finite eigenvalues of the pencil <math>sE-A</math> (<math>E = I</math>, if not otherwise specified) have negative real part; consequently, the solution of the corresponding LTI <math>with u(t) = 0</math> tends to 0 as <math>t \to \infty</math> 2 = ? 3 = ? (other stability definitions added and defined as necessary)
|- style="vertical-align:middle;"
| <code>isStateSpaceSymm</code>
| (bool) <math>A = A^T</math>, <math>C = B^T</math>, <math>E = E^T</math>, <math>D = D^T</math>; trivially implies <code>isSysSymm</code>
|- style="vertical-align:middle;"
| <code>isSysSymm</code>
| (bool) <math>H(s) = C (sE -A)^{-1} B+D</math> is symmetric; equivalently, the system's Markov parameters are symmetric For a distinction between symmetry and state-space symmetry, see Def 2.1 in [[https://www.sciencedirect.com/science/article/pii/S0167691198000243 this paper]]. [[https://doi.org/10.1109/PROC.1983.12688 Another useful source]]
|- style="vertical-align:middle;"
| <code>license</code>
| (char) license associated to data
|- style="vertical-align:middle;"
| LTI
| linear time-invariant
|- style="vertical-align:middle;"
| LTV
| linear time-varying
|- style="vertical-align:middle;"
| MIMO
| multiple-input, multiple-output
|- style="vertical-align:middle;"
| MOR
| model order reduction
|- style="vertical-align:middle;"
| <code>MORWikiLink</code>
| (char) link to MORWiki page
|- style="vertical-align:middle;"
| <code>MORWikiPageName</code>
| (char) fixed page name with plain text description of system in the MORWiki
|- style="vertical-align:middle;"
| <code>nInputs</code>
| (int) number of inputs
|- style="vertical-align:middle;"
| NL
| nonlinear; a nonlinear system depends nonlinearly on <math>x(t)</math>
|- style="vertical-align:middle;"
| <code>nnz*</code>
| (list of int) when the matrix * is sparse, the number of nonzeros; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>nOutputs</code>
| (int) number of outputs
|- style="vertical-align:middle;"
| <code>nParameters</code>
| (int) maximum number of parameters in a parametric system
|- style="vertical-align:middle;"
| <code>nStates</code>
| (int) number of states
|- style="vertical-align:middle;"
| <code>nUnstabPoles</code>
| (int) the number of poles with nonnegative real part, computed numerically
|- style="vertical-align:middle;"
| ODE
| ordinary differential equation
|- style="vertical-align:middle;"
| PDE
| partial differential equation
|- style="vertical-align:middle;"
| PMOR
| parametric model order reduction
|- style="vertical-align:middle;"
| POD
| proper orthogonal decomposition
|- style="vertical-align:middle;"
| QBS
| quadratic bilinear system
|- style="vertical-align:middle;"
| RBM
| reduced basis model
|- style="vertical-align:middle;"
| ROM
| reduced order model
|- style="vertical-align:middle;"
| SISO
| single-input, single-output
|- style="vertical-align:middle;"
| SOS
| second-order system
|- style="vertical-align:middle;"
| <code>systemClass</code>
| (char) model realization; see also [[Models]]
|- style="vertical-align:middle;"
| <code>zenodoDOI</code>
| (char) zenodo DOI (not the full URL!)
|}

Glossary

2023-05-09T15:54:38Z

Lund: fix typo

{{preliminary}} 

This page contains a glossary of terms related to the [[MORB]] collection, benchmark.csv, the [[Search Tool]], and this wiki as a whole.

{| class="wikitable" style="background-color:#FFF;"
|- style="font-weight:bold; vertical-align:middle;"
! Abbreviation / Attribute / Term
! Description
|- style="vertical-align:middle;"
| <code>_comments</code>
| (char) internal comments
|- style="vertical-align:middle;"
| <code>_dataSource</code>
| (char) where the data was pulled from during processing: [[https://gitlab.mpi-magdeburg.mpg.de/himpe/unimat unimat]], [[https://sites.google.com/site/rommes/software rommes]], or [[https://modelreduction.org morwiki]]
|- style="vertical-align:middle;"
| <code>_sourceFilename</code>
| (char) filename of data used during processing
|- style="vertical-align:middle;"
| AP
| affine parametric; parameters of any component are linear combinations of fixed matrices
|- style="vertical-align:middle;"
| <code>components</code>
| (char) matrices stored in .mat file; see also [[Models]]
|- style="vertical-align:middle;"
| <code>cond*</code>
| (list of float) the numerically computed ratio between the largest and smallest singular values of the matrix * (including 0); Inf is treated as a float in MATLAB; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>creator</code>
| (list of char) ORCID(s) of original author(s) or creator(s) of dataset
|- style="vertical-align:middle;"
| DAE
| differential algebraic equation; subset of LTI
|- style="vertical-align:middle;"
| <code>daeDiffIndex</code>
| (int) DAE differentiation index; roughly the number of differentiations required to make a system an ODE
|- style="vertical-align:middle;"
| DEIM
| discrete empirical interpolation method
|- style="vertical-align:middle;"
| <code>directory</code>
| (char) folder / collection to which a benchmark belongs
|- style="vertical-align:middle;"
| DSPMR
| dominant subspace projection model reduction
|- style="vertical-align:middle;"
| <code>editor</code>
| (list of char) ORCID(s) of editor(s) who converted data to new .mat format
|- style="vertical-align:middle;"
| <code>filename</code>
| (char) filename and benchmark ID
|- style="vertical-align:middle;"
| FOM
| full order model
|- style="vertical-align:middle;"
| FOS
| first-order system
|- style="vertical-align:middle;"
| <code>is*CholAble</code>
| (list of bool) the matrix * is symmetric positive definite, i.e., it admits a Cholesky factorization (and is therefore "Cholesky-able"); note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>is*Sparse</code>
| (list of bool) the matrix * is stored in a sparse format; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>is*Symm</code>
| (list of bool) the matrix is symmetric; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>isContractive</code>
| (bool) a system that is dissipative w.r.t. <math>s(u(t),y(t)) = ||u(t)||^2 - ||y(t)||^2</math>
|- style="vertical-align:middle;"
| <code>isDAE</code>
| (bool) differential algebraic equation; an LTI DAE has a nonzero, singular E component
|- style="vertical-align:middle;"
| <code>isPassive</code>
| (bool) a system that is dissipative w.r.t s(u(t),y(t)) = u(t)^T y(t)
|- style="vertical-align:middle;"
| <code>isSquare</code>
| (bool) nInputs = nOutputs
|- style="vertical-align:middle;"
| <code>isStable</code>
| (int) numerically computed; 0 = no stability 1 = asymptotically stable; i.e., all finite eigenvalues of the pencil <math>sE-A</math> (<math>E = I</math>, if not otherwise specified) have negative real part; consequently, the solution of the corresponding LTI <math>with u(t) = 0</math> tends to 0 as <math>t \to \infty</math> 2 = ? 3 = ? (other stability definitions added and defined as necessary)
|- style="vertical-align:middle;"
| <code>isStateSpaceSymm</code>
| (bool) <math>A = A^T</math>, <math>C = B^T</math>, <math>E = E^T</math>, <math>D = D^T</math>; trivially implies <code>isSysSymm</code>
|- style="vertical-align:middle;"
| <code>isSysSymm</code>
| (bool) <math>H(s) = C (sE -A)^{-1} B+D</math> is symmetric; equivalently, the system's Markov parameters are symmetric For a distinction between symmetry and state-space symmetry, see Def 2.1 in [[https://www.sciencedirect.com/science/article/pii/S0167691198000243 this paper]]. [[https://doi.org/10.1109/PROC.1983.12688 Another useful source]]
|- style="vertical-align:middle;"
| <code>license</code>
| (char) license associated to data
|- style="vertical-align:middle;"
| LTI
| linear time-invariant
|- style="vertical-align:middle;"
| LTV
| linear time-varying
|- style="vertical-align:middle;"
| MIMO
| multiple-input, multiple-output
|- style="vertical-align:middle;"
| MOR
| model order reduction
|- style="vertical-align:middle;"
| <code>MORWikiLink</code>
| (char) link to MORWiki page
|- style="vertical-align:middle;"
| <code>MORWikiPageName</code>
| (char) fixed page name with plain text description of system in the MORWiki
|- style="vertical-align:middle;"
| <code>nInputs</code>
| (int) number of inputs
|- style="vertical-align:middle;"
| NL
| nonlinear; a nonlinear system depends nonlinearly on <math>x(t)</math>
|- style="vertical-align:middle;"
| <code>nnz*</code>
| (list of int) when the matrix * is sparse, the number of nonzeros; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>nOutputs</code>
| (int) number of outputs
|- style="vertical-align:middle;"
| <code>nParameters</code>
| (int) maximum number of parameters in a parametric system
|- style="vertical-align:middle;"
| <code>nStates</code>
| (int) number of states
|- style="vertical-align:middle;"
| <code>nUnstabPoles</code>
| (int) the number of poles with nonnegative real part, computed numerically
|- style="vertical-align:middle;"
| ODE
| ordinary differential equation
|- style="vertical-align:middle;"
| PDE
| partial differential equation
|- style="vertical-align:middle;"
| PMOR
| parametric model order reduction
|- style="vertical-align:middle;"
| POD
| proper orthogonal decomposition
|- style="vertical-align:middle;"
| QBS
| quadratic bilinear system
|- style="vertical-align:middle;"
| RBM
| reduced basis model
|- style="vertical-align:middle;"
| ROM
| reduced order model
|- style="vertical-align:middle;"
| SISO
| single-input, single-output
|- style="vertical-align:middle;"
| SOS
| second-order system
|- style="vertical-align:middle;"
| <code>systemClass</code>
| (char) model realization; see also [[Models]]
|- style="vertical-align:middle;"
| <code>zenodoDOI</code>
| (char) zenodo DOI (not the full URL!)
|}

Glossary

2023-05-09T15:54:21Z

Lund: initialize glossary

{{preliminary}} 

This page contains a glossary of terms related to [[MORB]] collection, benchmark.csv, the [[Search Tool]], and this wiki as a whole.

{| class="wikitable" style="background-color:#FFF;"
|- style="font-weight:bold; vertical-align:middle;"
! Abbreviation / Attribute / Term
! Description
|- style="vertical-align:middle;"
| <code>_comments</code>
| (char) internal comments
|- style="vertical-align:middle;"
| <code>_dataSource</code>
| (char) where the data was pulled from during processing: [[https://gitlab.mpi-magdeburg.mpg.de/himpe/unimat unimat]], [[https://sites.google.com/site/rommes/software rommes]], or [[https://modelreduction.org morwiki]]
|- style="vertical-align:middle;"
| <code>_sourceFilename</code>
| (char) filename of data used during processing
|- style="vertical-align:middle;"
| AP
| affine parametric; parameters of any component are linear combinations of fixed matrices
|- style="vertical-align:middle;"
| <code>components</code>
| (char) matrices stored in .mat file; see also [[Models]]
|- style="vertical-align:middle;"
| <code>cond*</code>
| (list of float) the numerically computed ratio between the largest and smallest singular values of the matrix * (including 0); Inf is treated as a float in MATLAB; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>creator</code>
| (list of char) ORCID(s) of original author(s) or creator(s) of dataset
|- style="vertical-align:middle;"
| DAE
| differential algebraic equation; subset of LTI
|- style="vertical-align:middle;"
| <code>daeDiffIndex</code>
| (int) DAE differentiation index; roughly the number of differentiations required to make a system an ODE
|- style="vertical-align:middle;"
| DEIM
| discrete empirical interpolation method
|- style="vertical-align:middle;"
| <code>directory</code>
| (char) folder / collection to which a benchmark belongs
|- style="vertical-align:middle;"
| DSPMR
| dominant subspace projection model reduction
|- style="vertical-align:middle;"
| <code>editor</code>
| (list of char) ORCID(s) of editor(s) who converted data to new .mat format
|- style="vertical-align:middle;"
| <code>filename</code>
| (char) filename and benchmark ID
|- style="vertical-align:middle;"
| FOM
| full order model
|- style="vertical-align:middle;"
| FOS
| first-order system
|- style="vertical-align:middle;"
| <code>is*CholAble</code>
| (list of bool) the matrix * is symmetric positive definite, i.e., it admits a Cholesky factorization (and is therefore "Cholesky-able"); note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>is*Sparse</code>
| (list of bool) the matrix * is stored in a sparse format; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>is*Symm</code>
| (list of bool) the matrix is symmetric; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>isContractive</code>
| (bool) a system that is dissipative w.r.t. <math>s(u(t),y(t)) = ||u(t)||^2 - ||y(t)||^2</math>
|- style="vertical-align:middle;"
| <code>isDAE</code>
| (bool) differential algebraic equation; an LTI DAE has a nonzero, singular E component
|- style="vertical-align:middle;"
| <code>isPassive</code>
| (bool) a system that is dissipative w.r.t s(u(t),y(t)) = u(t)^T y(t)
|- style="vertical-align:middle;"
| <code>isSquare</code>
| (bool) nInputs = nOutputs
|- style="vertical-align:middle;"
| <code>isStable</code>
| (int) numerically computed; 0 = no stability 1 = asymptotically stable; i.e., all finite eigenvalues of the pencil <math>sE-A</math> (<math>E = I</math>, if not otherwise specified) have negative real part; consequently, the solution of the corresponding LTI <math>with u(t) = 0</math> tends to 0 as <math>t \to \infty</math> 2 = ? 3 = ? (other stability definitions added and defined as necessary)
|- style="vertical-align:middle;"
| <code>isStateSpaceSymm</code>
| (bool) <math>A = A^T</math>, <math>C = B^T</math>, <math>E = E^T</math>, <math>D = D^T</math>; trivially implies <code>isSysSymm</code>
|- style="vertical-align:middle;"
| <code>isSysSymm</code>
| (bool) <math>H(s) = C (sE -A)^{-1} B+D</math> is symmetric; equivalently, the system's Markov parameters are symmetric For a distinction between symmetry and state-space symmetry, see Def 2.1 in [[https://www.sciencedirect.com/science/article/pii/S0167691198000243 this paper]]. [[https://doi.org/10.1109/PROC.1983.12688 Another useful source]]
|- style="vertical-align:middle;"
| <code>license</code>
| (char) license associated to data
|- style="vertical-align:middle;"
| LTI
| linear time-invariant
|- style="vertical-align:middle;"
| LTV
| linear time-varying
|- style="vertical-align:middle;"
| MIMO
| multiple-input, multiple-output
|- style="vertical-align:middle;"
| MOR
| model order reduction
|- style="vertical-align:middle;"
| <code>MORWikiLink</code>
| (char) link to MORWiki page
|- style="vertical-align:middle;"
| <code>MORWikiPageName</code>
| (char) fixed page name with plain text description of system in the MORWiki
|- style="vertical-align:middle;"
| <code>nInputs</code>
| (int) number of inputs
|- style="vertical-align:middle;"
| NL
| nonlinear; a nonlinear system depends nonlinearly on <math>x(t)</math>
|- style="vertical-align:middle;"
| <code>nnz*</code>
| (list of int) when the matrix * is sparse, the number of nonzeros; note that AP systems will have a list of floats, with each entry pertaining to the matrix of the same index
|- style="vertical-align:middle;"
| <code>nOutputs</code>
| (int) number of outputs
|- style="vertical-align:middle;"
| <code>nParameters</code>
| (int) maximum number of parameters in a parametric system
|- style="vertical-align:middle;"
| <code>nStates</code>
| (int) number of states
|- style="vertical-align:middle;"
| <code>nUnstabPoles</code>
| (int) the number of poles with nonnegative real part, computed numerically
|- style="vertical-align:middle;"
| ODE
| ordinary differential equation
|- style="vertical-align:middle;"
| PDE
| partial differential equation
|- style="vertical-align:middle;"
| PMOR
| parametric model order reduction
|- style="vertical-align:middle;"
| POD
| proper orthogonal decomposition
|- style="vertical-align:middle;"
| QBS
| quadratic bilinear system
|- style="vertical-align:middle;"
| RBM
| reduced basis model
|- style="vertical-align:middle;"
| ROM
| reduced order model
|- style="vertical-align:middle;"
| SISO
| single-input, single-output
|- style="vertical-align:middle;"
| SOS
| second-order system
|- style="vertical-align:middle;"
| <code>systemClass</code>
| (char) model realization; see also [[Models]]
|- style="vertical-align:middle;"
| <code>zenodoDOI</code>
| (char) zenodo DOI (not the full URL!)
|}

Vertical Stand

2023-05-09T14:57:57Z

Lund: /* Data */ comment on re-indexing

[[Category:benchmark]]
[[Category:Parametric]]
[[Category:Sparse]]
[[Category:linear]]
[[Category:time varying]]
[[Category:first differential order]]
[[Category:ODE]]
[[Category:MIMO]]

__NUMBEREDHEADINGS__

==Description==

<figure id="fig:cad">
[[File:Staender_Geom_white.jpg|180px|thumb|right|<caption>CAD Geometry</caption>]]
</figure>

The '''vertical stand''' (see Fig. 1) represents a structural part of a machine tool. A pair of guide rails is located on one of the surfaces of this structural part,
and during the machining process, a tool slide is moved to different positions along these rails. The machining process produces a certain amount of heat which is transported through the slide structure into the '''vertical stand'''. This heat source is considered to be a temperature input <math>q_{th}(t)</math> at the guide rails. The induced temperature field, denoted by <math> T </math> is modeled by the [[wikipedia:heat equation|heat equation]]

:<math>
c_p\rho\frac{\partial{T}}{\partial{t}}=\Delta T=0
</math>

with the boundary conditions

:<math>
\lambda\frac{\partial T}{\partial n}=q_{th}(t) \qquad\qquad\qquad
</math> on <math> \Gamma_{rail} </math> (surface where the tool slide is moving on the guide rails),

describing the heat transfer between the tool slide and the '''vertical stand'''.
The heat transfer to the ambiance is given by the locally fixed [[wikipedia:Robin_boundary_condition|Robin-type boundary condition]]

:<math>
\lambda\frac{\partial T}{\partial n}=\kappa_i(T-T_i^{ext})
</math> on <math> \Gamma_{amb} </math> (remaining boundaries).

The motion driven temperature input and the associated change in the temperature field <math>T</math> lead to deformations <math>u</math> within the stand structure. Further, it is assumed that no external forces <math>q_{el}</math> are induced to the system, such that the deformation is purely driven by the change of temperature. Since the mechanical behavior of the machine stand is much faster than the propagation of the thermal field, it is sufficient to consider the stationary [[wikipedia:Linear_elasticity|linear elasticity]] equations
:<math>
\begin{align}
-\operatorname{div}(\sigma(u)) &=q_{el}=0&\text{ on }\Omega,\\
\varepsilon(u) &= {\mathbf{C}}^{-1}:\sigma(u)+\beta(T-T_{ref})I_d&\text{ on }\Omega,\\
{\mathbf{C}}^{-1}\sigma(u) &=\frac{1+\nu}{E_u}\sigma(u)-\frac{\nu}{E_u}\text{tr}(\sigma(u))I_d&\text{ on }\Omega,\\
\varepsilon(u) &= \frac{1}{2}(\nabla u+\nabla u^T)&\text{on }\Omega.
\end{align}
</math>

'''Geometrical dimensions:'''

Stand: Width (<math>x</math> direction): <math>519mm</math>, Height (<math>y</math> direction): <math>2\,010mm</math>, Depth (<math>z</math> direction): <math>480mm</math>

Guide rails: <math>y\in [519, 2\,004] mm</math>

Slide: Width: <math>430 mm</math>, Height: <math>500mm</math>, Depth: <math>490 mm</math>

===Discretized Model===
The solid model has been generated and meshed in [[wikipedia:ANSYS|ANSYS]].
For the spatial discretization the [[wikipedia:finite element method|finite element method]] with linear Lagrange elements has been used and is implemented in [[wikipedia:FEniCS Project|FEniCS]]. The resulting system of [[wikipedia:ordinary differential equations|ordinary differential equations (ODE)]], representing the thermal behavior of the stand, reads

:<math>
\begin{align}
E \frac{\partial}{\partial t} T(t) &= A(t)T(t) + B(t)z(t), \\
T(0) &= T_0,
\end{align}
</math>

with <math>t>0</math> and a system dimension of <math>n=16\,626</math> degrees of freedom and <math>m=6</math> inputs. Note that <math>A(.)\in\mathbb{R}^{n\times n}</math> and <math>B(.)\in\mathbb{R}^{n\times m}</math> are time-dependent matrix-valued functions. That is, the underlying model is represented by a linear time-Varying (LTV) state-space system. More precisely, here the time dependence originates from the change of the boundary condition on <math>\Gamma_{rail}</math> due to the motion of the tool slide. The system input is, according to the boundary conditions, given by
:<math>
z_i=\begin{cases}
q_{th}(t), i=1,\\
\kappa_i T_i^{ext}(t), i=2,\dots,6
\end{cases}.
</math>

The discretized stationary elasticity model becomes
:<math>
Mu(t)=KT(t).
</math>
For the observation of the displacements in single points/regions of interest an output equation of the form
:<math>
y(t) = C u(t)
</math>
is given.

Exploiting the one-sided coupling of the temperature and deformation fields, and reorganizing the elasticity equation in the form <math>u(t)=M^{-1}KT(t)</math>, the heat equation and the elasticity model can easily be combined via the output equation. Finally, the thermo-elastic control system is of the form
:<math>
\begin{align}
E \frac{\partial}{\partial t} T(t) &= A(t)T(t) + B(t)z(t), \\
y(t)& = \tilde{C}T(t),\\
T(0) &= T_0,
\end{align}
</math>
where the modified output matrix <math>\tilde{C}=CM^{-1}KT(t)</math> includes the entire elasticity information.

The motion of the tool slide and the associated variation of the affected input boundary are modeled by two different system representations. The following specific model representations have been developed and investigated in <ref name="morLanSB14" />, <ref name="LanSB15" />, <ref name="Lan17" />.

===Switched Linear System===
<figure id="fig:segm">
[[File:Slide_stand_scheme_new.pdf|thumb|right|220px|<caption>Schematic segmentation</caption>]]
</figure>

For the model description as a switched linear system, the guide rails of the machine stand are modeled as fifteen equally distributed horizontal segments with a height of <math>99mm</math> (see a schematic depiction in Fig. 2). Any of these segments is assumed to be completely covered by the tool slide if its midpoint (in y-direction) lies within the height of the slide. On the other hand, each segment whose midpoint is not covered is treated as not in contact and therefore the slide always covers five to six segments at each time. Still, the covering of six segments does not have a significant effect on the behavior of the temperature and displacement fields. Due to that and in order to keep the number of subsystems small, this scenario will be neglected. Then, in fact eleven distinct, discrete boundary condition configurations for the stand model that are prescribed by the geometrical dimensions of the segmentation and the tool slide are defined. These distinguishable setups define the subsystems
:<math>
\begin{align}
E\dot{T}(t)&=A_{\alpha}T(t)+B_{\alpha}z(t),\\
y(t)&=\tilde{C}T(t),
\end{align}
</math>
of the switched linear system <ref name=Lib03/>, where <math>\alpha</math> is a piecewise constant function of time, which takes its value from the index set <math>\mathcal{J}=\{1,\dots,11\}</math>. To be more precise, the switching signal <math>\alpha</math> implicitly maps the slide position to the number of the currently active subsystem.

===Linear Parameter-Varying System===
For the parametric model description, the finite element nodes located at the guide rails are clustered with respect to their y-coordinates. This results in <math>233</math> distinct layers in y-direction. According to these layers, the matrices <math>A(t)=A(\mu(t)), B(t)=B(\mu(t)) </math> are defined in a parameter-affine representation of the form
:<math>
\begin{align}
A(\mu) = A_0+f_1(\mu)A_1+...+f_{m_A}(\mu)A_{m_A},\\
B(\mu) = B_0+g_1(\mu)B_1+...+g_{m_B}(\mu)B_{m_B}
\end{align}
</math>
with the scalar functions <math>f_i,g_j\in\{0,1\}, i=1,...,m_A, j=1,...,m_B</math> selecting the active layers, covered by the tool slide and <math>\mu(t)</math> being the position of the middle point (vertical / y-direction) of the slide. The matrix <math>A_0\in\mathbb{R}^{n\times n}</math> consists of the discretization of the [[wikipedia:Laplace operator|Laplacian]] <math>\Delta</math>, as well as the discrete portions from the Robin-type boundaries that correspond to the temperature exchange with the ambiance. The remaining summands <math>A_j\in\mathbb{R}^{n\times n},~j=1,...,m_A</math> denote the discretization associated to the moving Robin-type boundaries. For the representation of the input matrix, the summand <math>B_0\in\mathbb{R}^{n\times 6}</math> consists of a single zero column followed by five columns related to the inputs <math>z_i,~i=2,...,6</math>. The remaing matrices <math>B_j,~j=1,...,m_B</math> are built by a single column corresponding to the different layers followed by a zero block of dimension <math>n\times 5</math> designated to fit the dimension of <math>B_0</math>.

Note that in general, the number of summands of these representations need not be equal. Still, according to the number of layers, for this example, it holds that <math>m_A=m_B=233</math>. For more details on parametric models, see e.g., <ref name="morBauBBetal11" /> and the references therein.

Then, the final linear parameter-varying (LPV) reformulation of the above LTV system reads
:<math>
\begin{align}
E\dot{T}(t)&=A(\mu)T(t)+B(\mu)z(t),\\
y(t)&=\tilde{C}T(t).
\end{align}
</math>

==Acknowledgement & Origin==
The base model was developed <ref name="GalGM11" />, <ref name="GalGM15" /> in the [http://transregio96.de Collaborative Research Centre Transregio 96] ''Thermo-Energetic Design of Machine Tools'' funded by the [http://www.dfg.de/en/index.jsp Deutsche Forschungsgemeinschaft] .

==Data==
====Switched System Data====
The data file [[Media:VertStand_SLS.tar.gz|VertStand_SLS.tar.gz]] contains the matrices


:<math>
\begin{align}
E\in\mathbb{R}^{n\times n}, \tilde{C}\in\mathbb{R}^{q\times n}, A_\alpha\in\mathbb{R}^{n\times n}, B_\alpha\in\mathbb{R}^{n\times m}, \alpha=1,\dots,11.
\end{align}
</math>
defining the subsystems of the switched linear system.
The matrices <math>A_\alpha</math> and <math>B_\alpha</math> are numbered according to the slide position in descending order (1 - uppermost slide position / 11 - lowest slide position).

System dimensions:
:<math>
n=16\,626, m=6, q=27
</math>

====Parametric System Data====
The data file [[Media:VertStand_PAR.tar.gz|VertStand_PAR.tar.gz]] contains the matrices

:<math>
E,A_j\in\mathbb{R}^{n\times n},j=1,...,234, B_{rail}\in\mathbb{R}^{n\times 233}, B_{amb}\in\mathbb{R}^{n\times 5}</math> and <math>\tilde{C}\in\mathbb{R}^{q\times n}, q=27, n=16\,626,
</math>
as well as a file ''ycoord_layers.mtx'' containing the y-coordinates of the layers located on the guide rails.

Here <math>B_{rail} </math> contains all columns corresponding to the different layers on the guide rails and <math>B_{amb}</math> correlates to the boundaries where the ambient temperatures act on.

In order to set up the parameter dependent matrices <math>A(\mu),B(\mu)</math> the active matrices <math>A_i</math> and columns <math>B_{rail}(:,i)</math> associated to the covered layers have to be identified by the current position <math>\mu(t)</math> (vertical middle point of the slide) and the geometrical dimensions of the tool slide and the y-coordinates of the different layers given in ''ycoord_layers.mtx''. Then, <math>B(\mu)</math> has to be set up in the form <math>B(\mu)=[\sum_{i\in id_{active}}\!\!\!\!\!B_{rail}(:,i),B_{amb}]</math>, where <math>id_{active}</math> denotes the set of covered layers and their corresponding columns in <math>B_{rail}</math>.



The matrices have been re-indexed (starting with 1) for [[MORB]].

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_vertstand,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Vertical Stand},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Vertical_Stand}</nowiki>,
year = 2014
}

* For the background on the benchmark:

@Article{morLanSB14,
author = {Lang, N. and Saak, J. and Benner, P.},
title = {Model Order Reduction for Systems with Moving Loads},
journal = {at-Automatisierungstechnik},
volume = 62,
number = 7,
pages = {512--522},
year = 2014,
doi = {10.1515/auto-2014-1095}
}

==References==

<references>

<ref name="morBauBBetal11"> U. Baur, C. A. Beattie, P. Benner, and S. Gugercin,
"[http://epubs.siam.org/doi/abs/10.1137/090776925 Interpolatory projection methods for parameterized model reduction]",
SIAM J. Sci. Comput., 33(5):2489-2518, 2011</ref>

<ref name="Lib03">D. Liberzon, [https://doi.org/10.1007/978-3-319-12625-8_8 Switching in Systems and Control] , Springer-Verlag, New York, 2003</ref>

<ref name="GalGM11"> A. Galant, K. Großmann, and A. Mühl, Model Order Reduction (MOR) for
Thermo-Elastic Models of Frame Structural Components on Machine Tools.
ANSYS Conference & 29th CADFEM Users’ Meeting 2011, October
19-21, 2011, Stuttgart, Germany</ref>.

<ref name="morLanSB14">N. Lang and J. Saak and P. Benner,
[https://doi.org/10.1515/auto-2014-1095 Model Order Reduction for Systems with Moving Loads] ,
in De Gruyter Oldenbourg: at-Automatisierungstechnik, Volume 62, Issue 7, Pages 512-522, 2014</ref>

<ref name="LanSB15">N. Lang, J. Saak and P. Benner, [https://doi.org/10.1007/978-3-319-12625-8_8 Model Order Reduction for Thermo-Elastic Assembly Group Models] , In: Thermo Energetic Design of Machine Tools, Lecture Notes in Production Engineering, 85-92, 2015</ref>

<ref name="Lan17">N. Lang, [https://www.logos-verlag.de/cgi-bin/buch/isbn/4700 Numerical Methods for Large-Scale Linear Time-Varying Control Systems and related Differential Matrix Equations] , Logos-Verlag, 2018. ISBN: 978-3-8325-4700-4</ref>

<ref name="GalGM15">A. Galant, K. Großmann and A. Mühl, [https://doi.org/10.1007/978-3-319-12625-8_7 Thermo-Elastic Simulation of Entire Machine Tool] , In: Thermo Energetic Design of Machine Tools, Lecture Notes in Production Engineering, 69-84, 2015</ref>

</references>

==Contact==

'' [[User:Saak]]''

Modified Gyroscope

2023-05-09T14:56:18Z

Lund: /* Dimensions */ missing _

[[Category:benchmark]]
[[Category:ODE]]
[[Category:linear]]
[[Category:Parametric]]
[[Category:affine parameter representation]]
[[Category:time invariant]]
[[Category:second differential order]]

==Description==
<figure id="fig:gyro">[[File:Gyroscope.jpg|300px|thumb|right|<caption>Schematic representation of the gyroscope.</caption>]]</figure>

The device is a [[wikipedia:Vibrating_structure_gyroscope#MEMS_gyroscopes|MEMS gyroscope]] based on the butterfly gyroscope <ref name="lienemann2004"/> developed at the [http://www.imego.com/ Imego institute] in Gothenburg,
Sweden (see also: [[Butterfly_Gyroscope|Butterfly Gyroscope]], where a non-parametrized model for the device is given).
A [[wikipedia:Gyroscope|gyroscope]] is a device used to measure angular rates in up to three axes.

The basic working principle of the '''gyroscope''' can be described as follows, see also <ref name="Moo07"/>.
Without applied external rotation, the paddles vibrate in phase with the function <math>z(t),</math> see Fig. 1.
Under the influence of an external rotation about the <math>x</math>-axis (drawn in red),
an additional force due to the Coriolis acceleration acts upon the paddles.
This force leads to an additional small out-of-phase vibration between two paddles on the same side of the bearing.
This out-of phase vibration is measured as the difference of the <math>z</math>-displacement of the nodes with the red dots.
Thus, measuring the displacement of two adjacent paddles, the rotation velocity can be ascertained.

==Motivation==

When planning for and making decisions on future improvements of the butterfly gyroscope, it is of importance to improve the efficiency of the gyro simulations. Repeated analysis 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 run time for repeated simulations.

==The Parametrized Model==

Two parameters are of special interest for the model.
The first one is the quantity that is to be sensed, the rotation velocity <math>\theta</math> around the <math>x</math>-axes.
The second parameter is the width of the bearing, <math>d</math>.
The parametrized system below is obtained by finite element discretization of the parametrized model (in the form of partial differential equations) for the '''gyroscope'''.
The details of constructing the parametrized system can be found in <ref name="Moo07"/>.
The system is of the following form:

:<math>
\begin{align}
M(d)\ddot{x}(t) +E(d,\theta)\dot{x}(t) +K(d)x(t) &= B, \\
y(t) &=Cx(t).
\end{align}
</math>

Here,

* <math>M(d)=M + dM_1\in \mathbb R^{n\times n}</math> is the mass matrix,
* <math>E(d,\theta)=\theta(E_1 + d E_2)\in \mathbb R^{n\times n}</math> is the damping matrix,
* <math>K(d)=K+(1/d)K_1+dK_2\in \mathbb R^{n\times n}</math> is the stiffness matrix,
* <math>B \in \mathbb R^{n \times 1}</math> is the load vector,
* <math>C \in \mathbb R^{1 \times n}</math> is the output matrix,
* <math>x \in \mathbb R^{n}</math> is the state vector,
* and <math>y \in \mathbb R </math> is the output response.

The quantity of interesting <math>y</math> of the system is <math>\delta z(t)</math>,
which is the difference of the displacement <math>z(t)</math> between the two red markings on the ''east'' side of the bearing (see <xr id="fig:gyro"/>).

The parameters of the system, <math>d</math> and <math>\theta</math>,
represent the width of the bearing(<math>d</math>) and the rotation velocity along the <math>x</math>-axis (<math>\theta</math>),
with the ranges: <math>\theta\in [10^{-7}, 10^{-5}]</math> and <math>d\in [1,2]</math>.

The device works in the frequency range <math>f \in [0.025, 0.25]</math>MHz and the degrees of freedom are <math>n = 17913</math>.

==Data==

The model is generated in ANSYS.
The system matrices <math>M, \, M_1, \, E_1, \, E_2, \, K, \, K_1, \,K_2</math>, and <math> B</math> are in the [http://math.nist.gov/MatrixMarket/ MatrixMarket format], and can be downloaded here: [[Media: Gyroscope_modi.tgz|Gyroscope_modi.tgz]].
The matrix <math>C</math> defines the output, which has zeros on all the entries, except on the 2315th entry, where the value is <math>-1</math>, and on the 5806th entry, the value is <math>1</math>;
in MATLAB notation, it is <tt>C(1,2315) = -1</tt> and <tt>C(1,5806) = 1</tt>.

The matrices have been reindexed for [[MORB]], as shown in the next section.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
(M_1 + d M_2)\ddot{x}(t) + \theta(E_1 + d E_2) \dot{x}(t) + (K_1 + d^{-1} K_2 + d K_3)x(t) &=& B \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>M_{1,2}, E_{1,2}, K_{1,2,3} \in \mathbb{R}^{17931 \times 17931}</math>,
<math>B \in \mathbb{R}^{17931 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 17931}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_modgyro,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Modified Gyroscope},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Modified_Gyroscope}</nowiki>,
year = 2018
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morMoo07 morMoo07] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morMoo07 BibTeX])

==References==

<references>

<ref name="lienemann2004">J. Lienemann, D. Billger, E. B. Rudnyi, A. Greiner, J. G. Korvink, "[http://www.nsti.org/procs/Nanotech2004v2/6/W58.01 MEMS Compact Modeling Meets Model Order Reduction: Examples of the Application of Arnoldi Methods to Microsystem Devices]", TechConnect Briefs (Technical Proceedings of the 2004 NSTI Nanotechnology Conference and Trade Show, Volume 2): 303--306, 2004.</ref>

<ref name="Moo07">C. Moosmann, "[http://www.freidok.uni-freiburg.de/volltexte/3971/ ParaMOR Model Order Reduction for parameterized MEMS applications]", PhD thesis, Department of Microsystems Engineering, University of Freiburg, Freiburg, Germany 2007.</ref>

</references>

==Contact==

'' [[User:Feng]] ''

Modified Gyroscope

2023-05-09T14:56:02Z

Lund: /* Dimensions */ re-index matrices

[[Category:benchmark]]
[[Category:ODE]]
[[Category:linear]]
[[Category:Parametric]]
[[Category:affine parameter representation]]
[[Category:time invariant]]
[[Category:second differential order]]

==Description==
<figure id="fig:gyro">[[File:Gyroscope.jpg|300px|thumb|right|<caption>Schematic representation of the gyroscope.</caption>]]</figure>

The device is a [[wikipedia:Vibrating_structure_gyroscope#MEMS_gyroscopes|MEMS gyroscope]] based on the butterfly gyroscope <ref name="lienemann2004"/> developed at the [http://www.imego.com/ Imego institute] in Gothenburg,
Sweden (see also: [[Butterfly_Gyroscope|Butterfly Gyroscope]], where a non-parametrized model for the device is given).
A [[wikipedia:Gyroscope|gyroscope]] is a device used to measure angular rates in up to three axes.

The basic working principle of the '''gyroscope''' can be described as follows, see also <ref name="Moo07"/>.
Without applied external rotation, the paddles vibrate in phase with the function <math>z(t),</math> see Fig. 1.
Under the influence of an external rotation about the <math>x</math>-axis (drawn in red),
an additional force due to the Coriolis acceleration acts upon the paddles.
This force leads to an additional small out-of-phase vibration between two paddles on the same side of the bearing.
This out-of phase vibration is measured as the difference of the <math>z</math>-displacement of the nodes with the red dots.
Thus, measuring the displacement of two adjacent paddles, the rotation velocity can be ascertained.

==Motivation==

When planning for and making decisions on future improvements of the butterfly gyroscope, it is of importance to improve the efficiency of the gyro simulations. Repeated analysis 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 run time for repeated simulations.

==The Parametrized Model==

Two parameters are of special interest for the model.
The first one is the quantity that is to be sensed, the rotation velocity <math>\theta</math> around the <math>x</math>-axes.
The second parameter is the width of the bearing, <math>d</math>.
The parametrized system below is obtained by finite element discretization of the parametrized model (in the form of partial differential equations) for the '''gyroscope'''.
The details of constructing the parametrized system can be found in <ref name="Moo07"/>.
The system is of the following form:

:<math>
\begin{align}
M(d)\ddot{x}(t) +E(d,\theta)\dot{x}(t) +K(d)x(t) &= B, \\
y(t) &=Cx(t).
\end{align}
</math>

Here,

* <math>M(d)=M + dM_1\in \mathbb R^{n\times n}</math> is the mass matrix,
* <math>E(d,\theta)=\theta(E_1 + d E_2)\in \mathbb R^{n\times n}</math> is the damping matrix,
* <math>K(d)=K+(1/d)K_1+dK_2\in \mathbb R^{n\times n}</math> is the stiffness matrix,
* <math>B \in \mathbb R^{n \times 1}</math> is the load vector,
* <math>C \in \mathbb R^{1 \times n}</math> is the output matrix,
* <math>x \in \mathbb R^{n}</math> is the state vector,
* and <math>y \in \mathbb R </math> is the output response.

The quantity of interesting <math>y</math> of the system is <math>\delta z(t)</math>,
which is the difference of the displacement <math>z(t)</math> between the two red markings on the ''east'' side of the bearing (see <xr id="fig:gyro"/>).

The parameters of the system, <math>d</math> and <math>\theta</math>,
represent the width of the bearing(<math>d</math>) and the rotation velocity along the <math>x</math>-axis (<math>\theta</math>),
with the ranges: <math>\theta\in [10^{-7}, 10^{-5}]</math> and <math>d\in [1,2]</math>.

The device works in the frequency range <math>f \in [0.025, 0.25]</math>MHz and the degrees of freedom are <math>n = 17913</math>.

==Data==

The model is generated in ANSYS.
The system matrices <math>M, \, M_1, \, E_1, \, E_2, \, K, \, K_1, \,K_2</math>, and <math> B</math> are in the [http://math.nist.gov/MatrixMarket/ MatrixMarket format], and can be downloaded here: [[Media: Gyroscope_modi.tgz|Gyroscope_modi.tgz]].
The matrix <math>C</math> defines the output, which has zeros on all the entries, except on the 2315th entry, where the value is <math>-1</math>, and on the 5806th entry, the value is <math>1</math>;
in MATLAB notation, it is <tt>C(1,2315) = -1</tt> and <tt>C(1,5806) = 1</tt>.

The matrices have been reindexed for [[MORB]], as shown in the next section.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
(M_1 + d M_2)\ddot{x}(t) + \theta(E_1 + d E_2) \dot{x}(t) + (K_1 + d^{-1} K_2 + d K_3)x(t) &=& B \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>M_{1,2}, E_{1,2}, K{1,2,3} \in \mathbb{R}^{17931 \times 17931}</math>,
<math>B \in \mathbb{R}^{17931 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 17931}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_modgyro,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Modified Gyroscope},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Modified_Gyroscope}</nowiki>,
year = 2018
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morMoo07 morMoo07] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morMoo07 BibTeX])

==References==

<references>

<ref name="lienemann2004">J. Lienemann, D. Billger, E. B. Rudnyi, A. Greiner, J. G. Korvink, "[http://www.nsti.org/procs/Nanotech2004v2/6/W58.01 MEMS Compact Modeling Meets Model Order Reduction: Examples of the Application of Arnoldi Methods to Microsystem Devices]", TechConnect Briefs (Technical Proceedings of the 2004 NSTI Nanotechnology Conference and Trade Show, Volume 2): 303--306, 2004.</ref>

<ref name="Moo07">C. Moosmann, "[http://www.freidok.uni-freiburg.de/volltexte/3971/ ParaMOR Model Order Reduction for parameterized MEMS applications]", PhD thesis, Department of Microsystems Engineering, University of Freiburg, Freiburg, Germany 2007.</ref>

</references>

==Contact==

'' [[User:Feng]] ''

Modified Gyroscope

2023-05-09T14:55:09Z

Lund: /* Data */ comment on re-indexing

[[Category:benchmark]]
[[Category:ODE]]
[[Category:linear]]
[[Category:Parametric]]
[[Category:affine parameter representation]]
[[Category:time invariant]]
[[Category:second differential order]]

==Description==
<figure id="fig:gyro">[[File:Gyroscope.jpg|300px|thumb|right|<caption>Schematic representation of the gyroscope.</caption>]]</figure>

The device is a [[wikipedia:Vibrating_structure_gyroscope#MEMS_gyroscopes|MEMS gyroscope]] based on the butterfly gyroscope <ref name="lienemann2004"/> developed at the [http://www.imego.com/ Imego institute] in Gothenburg,
Sweden (see also: [[Butterfly_Gyroscope|Butterfly Gyroscope]], where a non-parametrized model for the device is given).
A [[wikipedia:Gyroscope|gyroscope]] is a device used to measure angular rates in up to three axes.

The basic working principle of the '''gyroscope''' can be described as follows, see also <ref name="Moo07"/>.
Without applied external rotation, the paddles vibrate in phase with the function <math>z(t),</math> see Fig. 1.
Under the influence of an external rotation about the <math>x</math>-axis (drawn in red),
an additional force due to the Coriolis acceleration acts upon the paddles.
This force leads to an additional small out-of-phase vibration between two paddles on the same side of the bearing.
This out-of phase vibration is measured as the difference of the <math>z</math>-displacement of the nodes with the red dots.
Thus, measuring the displacement of two adjacent paddles, the rotation velocity can be ascertained.

==Motivation==

When planning for and making decisions on future improvements of the butterfly gyroscope, it is of importance to improve the efficiency of the gyro simulations. Repeated analysis 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 run time for repeated simulations.

==The Parametrized Model==

Two parameters are of special interest for the model.
The first one is the quantity that is to be sensed, the rotation velocity <math>\theta</math> around the <math>x</math>-axes.
The second parameter is the width of the bearing, <math>d</math>.
The parametrized system below is obtained by finite element discretization of the parametrized model (in the form of partial differential equations) for the '''gyroscope'''.
The details of constructing the parametrized system can be found in <ref name="Moo07"/>.
The system is of the following form:

:<math>
\begin{align}
M(d)\ddot{x}(t) +E(d,\theta)\dot{x}(t) +K(d)x(t) &= B, \\
y(t) &=Cx(t).
\end{align}
</math>

Here,

* <math>M(d)=M + dM_1\in \mathbb R^{n\times n}</math> is the mass matrix,
* <math>E(d,\theta)=\theta(E_1 + d E_2)\in \mathbb R^{n\times n}</math> is the damping matrix,
* <math>K(d)=K+(1/d)K_1+dK_2\in \mathbb R^{n\times n}</math> is the stiffness matrix,
* <math>B \in \mathbb R^{n \times 1}</math> is the load vector,
* <math>C \in \mathbb R^{1 \times n}</math> is the output matrix,
* <math>x \in \mathbb R^{n}</math> is the state vector,
* and <math>y \in \mathbb R </math> is the output response.

The quantity of interesting <math>y</math> of the system is <math>\delta z(t)</math>,
which is the difference of the displacement <math>z(t)</math> between the two red markings on the ''east'' side of the bearing (see <xr id="fig:gyro"/>).

The parameters of the system, <math>d</math> and <math>\theta</math>,
represent the width of the bearing(<math>d</math>) and the rotation velocity along the <math>x</math>-axis (<math>\theta</math>),
with the ranges: <math>\theta\in [10^{-7}, 10^{-5}]</math> and <math>d\in [1,2]</math>.

The device works in the frequency range <math>f \in [0.025, 0.25]</math>MHz and the degrees of freedom are <math>n = 17913</math>.

==Data==

The model is generated in ANSYS.
The system matrices <math>M, \, M_1, \, E_1, \, E_2, \, K, \, K_1, \,K_2</math>, and <math> B</math> are in the [http://math.nist.gov/MatrixMarket/ MatrixMarket format], and can be downloaded here: [[Media: Gyroscope_modi.tgz|Gyroscope_modi.tgz]].
The matrix <math>C</math> defines the output, which has zeros on all the entries, except on the 2315th entry, where the value is <math>-1</math>, and on the 5806th entry, the value is <math>1</math>;
in MATLAB notation, it is <tt>C(1,2315) = -1</tt> and <tt>C(1,5806) = 1</tt>.

The matrices have been reindexed for [[MORB]], as shown in the next section.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
(M + d M_1)\ddot{x}(t) + \theta(E_1 + d E_2) \dot{x}(t) + (K + d^{-1} K_1 + d K_2)x(t) &=& B \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>M, M_1, E_1, E_2, K, K_1, K_2 \in \mathbb{R}^{17931 \times 17931}</math>,
<math>B \in \mathbb{R}^{17931 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 17931}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_modgyro,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Modified Gyroscope},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Modified_Gyroscope}</nowiki>,
year = 2018
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morMoo07 morMoo07] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morMoo07 BibTeX])

==References==

<references>

<ref name="lienemann2004">J. Lienemann, D. Billger, E. B. Rudnyi, A. Greiner, J. G. Korvink, "[http://www.nsti.org/procs/Nanotech2004v2/6/W58.01 MEMS Compact Modeling Meets Model Order Reduction: Examples of the Application of Arnoldi Methods to Microsystem Devices]", TechConnect Briefs (Technical Proceedings of the 2004 NSTI Nanotechnology Conference and Trade Show, Volume 2): 303--306, 2004.</ref>

<ref name="Moo07">C. Moosmann, "[http://www.freidok.uni-freiburg.de/volltexte/3971/ ParaMOR Model Order Reduction for parameterized MEMS applications]", PhD thesis, Department of Microsystems Engineering, University of Freiburg, Freiburg, Germany 2007.</ref>

</references>

==Contact==

'' [[User:Feng]] ''

Thermal Block

2023-05-09T14:44:06Z

Lund: /* Dimensions */ re-index

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:first differential order]]
[[Category:time invariant]]
[[Category:MIMO]]
[[Category:Sparse]]

==Description==
A parametric semi-discretized heat transfer problem with varying heat transfer coefficients, the parameters, on subdomains. This model is also called the ''cookie baking problem'', and can be viewed as a flattened 2-D version of the ''skyscraper problem'' from high-performance computing.

<figure id="fig1">[[File:ThermalBlockDomain.svg|490px|thumb|right|<caption>The computational domain and boundaries.</caption>]]</figure>
<figure id="fig1">[[File:ThermalBlockTend.png|490px|thumb|right|<caption>A sample heat distribution at time 1.0 for parameter choice [100, 0.01, 0.001, 0.0001].</caption>]]</figure>
<figure id="fig1">[[File:ThermalBlockSigmaMagnitude.png|490px|thumb|right|<caption>Sigma magnitude plot of the single parameter variant.</caption>]]</figure>

===Modeling===
Consider a parameter <math>\mu\in{[10^{-6},10^2]}^4\subset\mathbb{R}^{4}</math> and define the heat conductivity <math>\sigma(\xi; \mu)</math> as <math>1.0</math> when <math>\xi\in\Omega_0</math> and <math>\sigma(\xi; \mu)=\mu_i</math> when <math>\xi\in\Omega_i</math>. The heat distribution is governed by the equation:
:<math>
\partial_t \theta(t, \xi; \mu) + \nabla \cdot (- \sigma(\xi; \mu) \nabla \theta(t, \xi; \mu)) = 0,\text{ for } t\in (0,T), \text{ and } \xi \in \Omega,
</math>
with a heat-inflow condition on the left ([[wikipedia:Neumann_boundary_condition|Neumann boundary]])
:<math>
\sigma(\xi; \mu) \nabla \theta(t, \xi; \mu) \cdot n(\xi) = u(t)\text{ for } t \in (0,T), \text{ and } \xi \in \Gamma_{in},
</math>
perfect isolation on the top and bottom ([[wikipedia:Boundary_conditions_in_fluid_dynamics#Wall_boundary_condition|Neumann-zero boundary]])
:<math>
\sigma(\xi; \mu) \nabla \theta(t, \xi; \mu) \cdot n(\xi) = 0\text{ for } t \in (0,T), \text{ and } \xi \in \Gamma_N,
</math>
and fixed temperature on the right ([[wikipedia:Dirichlet_boundary_condition|Dirichlet boundary]])
:<math>
\theta(t, \xi; \mu) = 0\text{ for } t \in (0,T), \text{ and } \xi \in \Gamma_{D},
</math>
and initial condition
:<math>
\theta(0, \xi; \mu) = 0 \text{ for } \xi \in \Omega.
</math>

===Discretization===
For the discretization, [https://fenicsproject.org/ FEniCS] '''2019.1''' was used on a simplicial grid with first order elements. The mesh is generated from the domain specification using [http://gmsh.info/ gmsh] '''3.0.6''' with '<code>clscale</code>' set to <math>0.1</math>. The Python-based source code for the discretization can be found at [https://doi.org/10.5281/zenodo.3691894 Zenodo].

==Origin==
This benchmark was developed for the [https://imsc.uni-graz.at/modred2019/ MODRED 2019] proceedings<ref name="morRavS20/>.

==Data==
The benchmark includes the basic domain description as a gmsh input file, Python scripts for the matrix assembly, simulation in pyMOR, and visualization as VTK, together with the matrices both as one combined file <code>ABCE.mat</code> or separate matrix market files for all matrices. The sources and the <code>ABCE.mat</code> are available for download at [https://doi.org/10.5281/zenodo.3691894 Zenodo].

Note that the heat transfer coefficients are designed as characteristic functions on the domains, such that the system is only well-posed when all entries in <math>\mu</math> are positive.

==Dimensions==
System structure:
:<math>
\begin{align}
E\dot{x}(t) &= (A_1 + \mu_1 A_2 + \mu_2 A_3 + \mu_3 A_4 + \mu_4 A_5) x(t) + Bu(t), \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>E \in \mathbb{R}^{N \times N}</math>,
<math>A_{1,...,5} \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times 1}</math>,
<math>C \in \mathbb{R}^{4 \times N}</math>,

where <math>N=7\,488</math> for the system matrices given in <code>ABCE.mat</code>.

==Variants==
Besides the full four parameter setup, the model can be used in variations with other numbers of independent parameters. The following two are recommended in the original work and have been investigated in the literature<ref name="morBenW20c"/>,<ref name="morHim20"/>,<ref name="morBenKS20"/>,<ref name="morMliRS20"/>.

===Single parameter===
The interpretation of the thermal block as the "cookie baking" problem with slight variation in the dough leads to an easy one parameter variant. Here the new single parameter <math>\hat\mu\in [ 10^{-6}, 10^2]</math> is chosen such that <math> \mu = \hat\mu\left[0.2, 0.4, 0.6, 0.8\right]. </math>

===Non-parametric===
The system can be used as a standard LTI state-space model. It is suggested to use <math>\mu = \sqrt{10} [0.2, 0.4, 0.6, 0.8]</math>.

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

* For the benchmark itself and its data:
:: S. Rave and J. Saak, '''Thermal Block'''. MORwiki - Model Order Reduction Wiki, 2020. http://modelreduction.org/index.php/Thermal_Block

@MISC{morwiki_thermalblock,
author = <nowiki>{Rave, S. and Saak, J.}</nowiki>,
title = {Thermal Block},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Thermal_Block}</nowiki>,
year = 2020
}

* For the background on the benchmark:
:: S. Rave and J. Saak, [https://arxiv.org/abs/2003.00846 '''A Non-Stationary Thermal-Block Benchmark Model for Parametric Model Order Reduction''']. e-prints 2003.00846, arXiv, math.NA (2020).

@INPROCEEDINGS{morRavS21,
author = {Rave, S. and Saak, J.},
title = {A Non-Stationary Thermal-Block Benchmark Model for Parametric Model Order Reduction},
booktitle = {Model Reduction of Complex Dynamical Systems},
series = {International Series of Numerical Mathematics},
volume = {171},
publisher = {Springer},
year = 2021,
doi = {10.1007/978-3-030-72983-7_16}
}

==References==
<references>
<ref name="morBenW20c">P. Benner, S. W. R. Werner, [https://doi.org/10.1007/978-3-030-72983-7_19 MORLAB -- the Model Order Reduction LABoratory],
Model Reduction of Complex Dynamical Systems, International Series of Numerical Mathematics 171: 393--415, 2021.</ref>
<ref name="morHim20">C. Himpe, [https://doi.org/10.1007/978-3-030-72983-7_7 Comparing (empirical-Gramian-based) model order reduction algorithms],
Model Reduction of Complex Dynamical Systems, International Series of Numerical Mathematics 171: 141--164, 2021.</ref>
<ref name="morBenKS20">P. Benner, M. Köhler, J. Saak, [https://doi.org/10.1007/978-3-030-72983-7_18 Matrix equations, sparse solvers: M-M.E.S.S.-2.0.1 – philosophy, features and application for (parametric) model order reduction], Model Reduction of Complex Dynamical Systems, International Series of Numerical Mathematics 171: 369--392, 2021.</ref>
<ref name="morRavS20">S. Rave, J. Saak, [https://doi.org/10.1007/978-3-030-72983-7_16 An Non-Stationary Thermal-Block Benchmark Model for Parametric Model Order Reduction], Model Reduction of Complex Dynamical Systems, International Series of Numerical Mathematics 171: 349-356, 2021.</ref>
<ref name="morMliRS20">P. Mlinarić, S. Rave, J. Saak, [https://doi.org/10.1007/978-3-030-72983-7_17 Parametric model order reduction using pyMOR], Model Reduction of Complex Dynamical Systems, International Series of Numerical Mathematics 171: 357--367 , 2021.</ref>
</references>

==Contact==
'' [[User:Saak]]''

Silicon Nitride Membrane

2023-05-09T14:42:14Z

Lund: /* Dimensions */ re-index

[[Category:benchmark]]
[[Category:Parametric]]
[[Category:linear]]
[[Category:first differential order]]
[[Category:time invariant]]

==Description==

<figure id="fig:tempprof">[[File:Fig_1.png|right|frame|<caption>silicon nitride membrane temperature profile</caption>]]</figure>

A '''silicon nitride membrane''' ([[wikipedia:Silicon_nitride|SiN]] membrane) <ref name="bechthold10"/> can be a part of a gas sensor,
but also a part of an infra-red sensor, a microthruster, an optical filter etc.
This structure resembles a microhotplate similar to other micro-fabricated devices such as gas sensors <ref name="spannhake05"/> and infrared sources <ref name="graf04"/> (See also [[Gas_Sensor|Gas Sensor Benchmark]]).
See Fig. 1, the temperature profile for the SiN membrane.

The governing heat transfer equation in the membrane is:

:<math> \nabla \cdot (\kappa \nabla T)+Q - \rho c_p \cdot \frac{\partial T}{\partial t}=0 </math>

where <math>\kappa </math> is the thermal conductivity in <math>W m^{-1} K^{-1}</math>,
<math>c_p</math> is the specific heat capacity in <math>J kg^{-1} K^{-1}</math>,
<math>\rho</math> is the mass density in <math>kg m^{-3}</math> and <math>T</math> is the temperature distribution.
We assume a homogeneous heat generation rate over a lumped resistor:

:<math>Q = \frac{u^2(t)}{R(T)}</math>

with <math>Q</math> the heat generation rate per unit volume in <math>W m^{-3}</math>.
We use the initial condition <math>T_0 = 273K</math>,
and the Dirichlet boundary condition <math>T = 273 K</math> at the bottom of the computational domain.

The convection boundary condition at the top of the membrane is

:<math>
q=h(T-T_{air}),
</math>

where <math>h</math> is the heat transfer coefficient between the membrane and the ambient air in <math>W m^{-2} K^{-1}</math>.

==Discretization==

Under the above convection boundary condition and assuming <math>T_{air}=0</math>,
a finite element discretization of the heat transfer model leads to the parametrized system as below,

:<math>
(E_0+\rho c_p \cdot E_1) \dot{T} +(A_0 +\kappa \cdot A_1 +h \cdot A_2)T = B \frac{u^2(t)}{R(T)}, \quad
y=C T,
</math>

where the volumetric heat capacity <math>\rho c_p</math>,
thermal conductivity <math>\kappa</math> and the heat transfer coefficient <math>h</math> between the membrane are kept as parameters.
The volumetric hear capacity <math>\rho c_p</math> is the product of two independent variables,
i.e. the specific hear capacity <math>c_p</math> and the density <math>\rho</math>.
The range of interest for the four independent variables are <math>\kappa \in [2, 5]</math>, <math>c_p \in [400, 750]</math>, <math>\rho \in [3000,3200]</math> and <math> h \in [10, 12]</math> respectively.
The frequency range is <math>f \in [0,25]Hz</math>.
What is of interest is the output in time domain.
The interesting time interval is <math>t \in [0,0.04]s</math>.

Here <math>R(T)</math> is either a constant heat resistivity <math>R(T)=R_0</math>, or <math>R(T)=R_0(1+\alpha T)</math>,
which depends linearly on the temperature.
Here we use <math>R_0=274.94 \Omega</math> and temperature coefficient <math>\alpha=2.293 \pm 0.006 \times 10^{-4}</math>.
The model was created and meshed in ANSYS. It contains a constant load vector <math>B</math> corresponding to the constant input power of <math>2.49mW</math>.
The number of degrees of freedom is <math>n=60,020</math>.

The input function <math>u(t)</math> is a step function with the value <math>1</math>,
which disappears at the time <math>0.02s</math>.
This means between <math>0s</math> and <math>0.02s</math> input is one and after that it is zero.
However, be aware that <math>u(t)</math> is just a factor with which the load vector B is multiplied and which corresponds to the heating power of <math>2.49mW</math>.
This means if one keeps <math>u(t)</math> as suggested above, the device is heated with <math>2.49mW</math> for the time length of 0.02s and after that the heating is turned off.
If for whatever reason, one wants the heating power to be <math>5mW</math>, then <math>u(t)</math> has to be set equal to two, etc.
When <math>R(T)=R_0(1+\alpha T)</math>, it is a function of the state vector <math>T</math> and hence, the system has non-linear input (It is also called a weakly nonlinear system.).

==Data==

The model is generated in ANSYS. The system matrices are in [http://math.nist.gov/MatrixMarket/ MatrixMarket] format and can be downloaded here: [[Media: SiN_membrane.tgz|SiN_membrane.tgz]].

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
(E_1 + \rho c_p E_2)\dot{x}(t) &=& -(A_1 + \kappa A_2 + h A_3)x(t) + Bu(t) \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>E_{1,2} \in \mathbb{R}^{60020 \times 60020}</math>,
<math>A_{1,2,3} \in \mathbb{R}^{60020 \times 60020}</math>,
<math>B \in \mathbb{R}^{60020 \times 1}</math>,
<math>C \in \mathbb{R}^{2 \times 60020}</math>.

==References==

<references>

<ref name="bechthold10">T. Bechtold, D. Hohfeld, E. B. Rudnyi and M. Guenther, "[https://doi.org/10.1088/0960-1317/20/4/045030 Efficient extraction of thin-film thermal parameters from numerical models via parametric model order reduction]", J. Micromech. Microeng. 20(4): 045030, 2010.</ref>

<ref name="spannhake05">J. Spannhake, O. Schulz, A. Helwig, G. Müller and T. Doll, "[https://doi.org/10.1109/ICSENS.2005.1597811 Design, development and operational concept of an advanced MEMS IR source for miniaturized gas sensor systems]", Proc. Sensors: 762--765, 2005.</ref>

<ref name="graf04">M. Graf, D. Barrettino, S. Taschini, C. Hagleitner, A. Hierlemann and H. Baltes, "[https://doi.org/10.1021/ac035432h Metal oxide-based monolithic complementary metal oxide semiconductor gas sensor microsystem]", Anal. Chem., 76(15): 4437--4445, 2004.</ref>

</references>

==Contact==
'' [[User:Feng|Lihong Feng]] ''

[http://www.igs.uni-rostock.de/mitarbeiter/tamara-bechtold/ Tamara Bechtold]

Scanning Electrochemical Microscopy

2023-05-09T14:40:53Z

Lund: /* Data */ add note about matrix renaming

[[Category:benchmark]]
[[Category:parametric]]
[[Category:linear]]
[[Category:time varying]]
[[Category:Parametric]]
[[Category:first differential order]]
[[Category:nonzero initial condition]]

==Description==

<figure id="fig:cylin">
[[Image:Fig.1.JPG|thumb|right|300px|<caption>Cylindrical Electrode</caption>]]
</figure>

'''Scanning Electrochemical Microscopy''' (SECM) has many applications in current problems in the biological field.
Quantitative mathematical models have been developed for different operating modes of the SECM.
Except for some very specific problems, like the diffusion-controlled current on a circular electrode far away from the border,
solutions can only be obtained by numerical simulation, which is based on discretization of the model in space by an appropriate method like finite differences, finite elements, or boundary elements.
After discretization, a high-dimensional system of ordinary differential equations is obtained.
Its high dimensionality leads to high computational cost.

We consider a cylindrical electrode in Fig. 1.
The computation domain under the 2D-axisymmetric approximation includes the electrolyte under the electrode.
We assume that the concentration does not depend on the rotation angle.
A single chemical reaction takes place on the electrode:

:<math>Ox+e^-\Leftrightarrow Red, \quad \quad \quad \quad (1) </math>

where <math>Ox</math> and <math>Red</math> are two different species in the reaction.
According to the theory of SECM <ref name="mirkin01"/>, the species transport in the electrolyte is described by diffusion only.
The diffusion partial differential equation is given by the second [[wikipedia:Fick's_laws_of_diffusion|Fick's law]] as follows

:<math>
\frac{dc_1}{dt}=D_1\cdot \Delta ^2c_1 , \quad
\frac{dc_2}{dt}=D_2\cdot \Delta ^2c_2,
</math>

where <math>c_1</math> and <math>c_2</math> are the concentration fields of species <math>Ox</math> and <math>Red</math>, respectively.
The initial conditions are <math>c_1(0)=c_{1,0}</math> and <math>c_2(0)=c_{2,0}.</math> conditions at the glass and the bottom of the bath
are described by the Neumann boundary conditions of zero flux <math>\nabla c_1\cdot \vec{n}=0</math> and <math>\nabla c_2\cdot \vec{n}=0</math>.
Conditions at the border of the bulk are described by Dirichlet boundary conditions of constant concentration, equal to the initial conditions <math>c_1=c_{1,0}</math> and <math>c_2=c_{2,0}</math>.
The boundary conditions at the electrode are described by

:<math>
\nabla c_1\cdot \vec{n}=j, \,
\nabla c_2\cdot \vec{n}=-j. \quad \quad \quad \quad (2)
</math>

Here <math>j</math> is related to the forward reaction rate <math>k_f</math> and the backward reaction rate <math>k_b</math> through the [[wikipedia:Butler–Volmer_equation|Butler-Volmer equation]],

:<math>
j=k_f \cdot c_1-k_b \cdot c_2.
</math>

The reaction rates <math>k_f</math> and <math>k_b</math> are in the following form,

:<math>
k_f=k^0\exp{(\frac{\alpha z F(v(t)-v^0)}{RT})}, </math>

:<math>k_b=k^0\exp{(\frac{-(1-\alpha) z F(v(t)-v^0)}{RT})} .
</math>

Here, <math>k^0</math> is the heterogeneous standard rate constant, which is an empirical transmission factor for a heterogeneous reaction.
<math>F</math> is the [[wikipedia:Faraday_constant|Faraday-constant]], <math>R</math> is the [[wikipedia:Gas_constant|gas constant]], <math>T</math> is the temperature, and <math>z</math> is the number of exchanged electrons per reaction.
<math>u(t)=v(t)-v^0</math> is the difference between the electrode potential and the reference potential.
This difference, to which we refer below as voltage, changes during the measurement of a [[wikipedia:Voltammetry|voltammogram]].

==Model==

The control volume method has been used for the spatial discretization of (1).
Together with the boundary conditions, the resulting system of ordinary differential equations is as follows,

:<math>
E\frac{d\vec{c}}{dt}+K(u(t))\vec{c}-A\vec{c}=B,\quad
y(t)=C\vec{c},\quad
\vec{c}(0)=\vec{c}_0 \neq 0,
</math>

where E and <math>K(u(t))</math> are system matrices, <math>K(u(t))</math> is a function of voltage that in turn depends on time.
The voltage appears in the system matrix due to the boundary conditions (2).
The vector <math>\vec{c} \in \mathbb{R}^n</math> is the vector of unknown concentrations, which includes both the <math>Ox</math> and <math>Red</math> species.
The vector <math>B</math> is the load vector, which arises as a consequence of the Dirichlet boundary conditions imposed at the bulk boundary of the electrolyte. The total current is computed as an integral (sum) over the electrode surface.
The matrix <math>K(u(t))</math> has the following form,

:<math>
K(u(t))=K_1(u(t))+K_2(u(t)),
</math>

where <math>K_i(u(t))=h_i D_i, \, i=1,2,</math> and <math>h_1=\exp(\beta u(t)), \, h_2=\exp(-\beta u(t))</math>.
The voltage <math>u(t)</math> is a function of <math>\sigma</math>,

:<math>
u(t)=\sigma t-1, \, \text{for } t \leq \frac{2}{ \sigma}, \quad
u(t)=-\sigma t+3, \, \text{for } \frac{2}{ \sigma} < t \leq \frac{4}{ \sigma},
</math>

where <math>\sigma</math> can take four different values, <math>\sigma=0.5, \, 0.05, \, 0.005, \, 0.0005</math>. The constant <math>\beta</math> is computed from the parameters <math>\alpha, \, z, \, F, \, R,</math> and <math>T,</math> leading to the value <math>\beta=21.243036728240824</math>.
Although the system is a time-varying system, it can be considered as a parametrized system with two parameters <math>h_1</math> and <math>h_2</math>.

==Data==

The data of the system matrices <math>E, \ D_1, \ D_2, \ A, \ B, C</math> as well as the initial state <math>\vec{c}_0=x_0</math> are in MatrixMarket format (http://math.nist.gov/MatrixMarket/), and can be downloaded here [[Media:SECM.TGZ|SECM.tgz]]. The quantity of interest is the current which is computed by <math>I(t)=C(5,:)\vec{c}</math> in MATLAB notation. The associated plot is called the [[wikipedia:Cyclic_voltammetry|cyclic voltammogram]] <ref name="feng06"/>, which is the plot of the current changing with the voltage <math>u(t)</math>.

For the MOR Benchmark tool ([[MORB]]), the matrices <math>A, D_1, D_2</math> have been renamed <math>A_1, A_2, A_3</math>, respectively.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
E\dot{c}(t) &=& (A_1 - h_1 A_2 - h_2 A_3)c(t) + Bu(t) \\
y(t) &=& Cc(t)
\end{array}
</math>

System dimensions:

<math>E \in \mathbb{R}^{16\,912 \times 16\,912}</math>,
<math>A_{1,2,3} \in \mathbb{R}^{16\,912 \times 16\,912}</math>,
<math>B \in \mathbb{R}^{16\,912 \times 1}</math>,
<math>C \in \mathbb{R}^{5 \times 16\,912}</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Scanning Electrochemical Microscopy'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Scanning_Electrochemical_Microscopy

@MISC{morwiki_secm,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Scanning Electrochemical Microscopy},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Scanning_Electrochemical_Microscopy}</nowiki>,
year = {2018}
}

* For the background on the benchmark:

@ARTICLE{morFenKRetal06,
author = {L. Feng, D. Koziol, E.B. Rudnyi, and J.G. Korvink},
title = {Parametric Model Reduction for Fast Simulation of Cyclic Voltammograms},
journal = {Sensor Letters},
volume = 4,
number = 2,
pages = {165--173},
year = 2006,
doi = {10.1166/sl.2006.021}
}

==References==

<references>

<ref name="mirkin01"> M.V. Mirkin, "[https://doi.org/10.1201/b11850-7 Chapter 5: Theory]", In: A.J. Bard and M.V. Mirkin, (eds.), Scanning Electrochemical Microscopy, CRC Press: 144--199, 2001.</ref>

<ref name="feng06"> L. Feng, D. Koziol, E.B. Rudnyi, and J.G. Korvink, "[https://doi.org/10.1166/sl.2006.021 Parametric Model Reduction for Fast Simulation of Cyclic Voltammograms]", Sensor Letters, 4(2): 165--173, 2006.</ref>

</references>

==Contact==

'' [[User:Feng|Lihong Feng]] ''

Scanning Electrochemical Microscopy

2023-05-09T14:37:45Z

Lund: /* Dimensions */ update to match MORB

[[Category:benchmark]]
[[Category:parametric]]
[[Category:linear]]
[[Category:time varying]]
[[Category:Parametric]]
[[Category:first differential order]]
[[Category:nonzero initial condition]]

==Description==

<figure id="fig:cylin">
[[Image:Fig.1.JPG|thumb|right|300px|<caption>Cylindrical Electrode</caption>]]
</figure>

'''Scanning Electrochemical Microscopy''' (SECM) has many applications in current problems in the biological field.
Quantitative mathematical models have been developed for different operating modes of the SECM.
Except for some very specific problems, like the diffusion-controlled current on a circular electrode far away from the border,
solutions can only be obtained by numerical simulation, which is based on discretization of the model in space by an appropriate method like finite differences, finite elements, or boundary elements.
After discretization, a high-dimensional system of ordinary differential equations is obtained.
Its high dimensionality leads to high computational cost.

We consider a cylindrical electrode in Fig. 1.
The computation domain under the 2D-axisymmetric approximation includes the electrolyte under the electrode.
We assume that the concentration does not depend on the rotation angle.
A single chemical reaction takes place on the electrode:

:<math>Ox+e^-\Leftrightarrow Red, \quad \quad \quad \quad (1) </math>

where <math>Ox</math> and <math>Red</math> are two different species in the reaction.
According to the theory of SECM <ref name="mirkin01"/>, the species transport in the electrolyte is described by diffusion only.
The diffusion partial differential equation is given by the second [[wikipedia:Fick's_laws_of_diffusion|Fick's law]] as follows

:<math>
\frac{dc_1}{dt}=D_1\cdot \Delta ^2c_1 , \quad
\frac{dc_2}{dt}=D_2\cdot \Delta ^2c_2,
</math>

where <math>c_1</math> and <math>c_2</math> are the concentration fields of species <math>Ox</math> and <math>Red</math>, respectively.
The initial conditions are <math>c_1(0)=c_{1,0}</math> and <math>c_2(0)=c_{2,0}.</math> conditions at the glass and the bottom of the bath
are described by the Neumann boundary conditions of zero flux <math>\nabla c_1\cdot \vec{n}=0</math> and <math>\nabla c_2\cdot \vec{n}=0</math>.
Conditions at the border of the bulk are described by Dirichlet boundary conditions of constant concentration, equal to the initial conditions <math>c_1=c_{1,0}</math> and <math>c_2=c_{2,0}</math>.
The boundary conditions at the electrode are described by

:<math>
\nabla c_1\cdot \vec{n}=j, \,
\nabla c_2\cdot \vec{n}=-j. \quad \quad \quad \quad (2)
</math>

Here <math>j</math> is related to the forward reaction rate <math>k_f</math> and the backward reaction rate <math>k_b</math> through the [[wikipedia:Butler–Volmer_equation|Butler-Volmer equation]],

:<math>
j=k_f \cdot c_1-k_b \cdot c_2.
</math>

The reaction rates <math>k_f</math> and <math>k_b</math> are in the following form,

:<math>
k_f=k^0\exp{(\frac{\alpha z F(v(t)-v^0)}{RT})}, </math>

:<math>k_b=k^0\exp{(\frac{-(1-\alpha) z F(v(t)-v^0)}{RT})} .
</math>

Here, <math>k^0</math> is the heterogeneous standard rate constant, which is an empirical transmission factor for a heterogeneous reaction.
<math>F</math> is the [[wikipedia:Faraday_constant|Faraday-constant]], <math>R</math> is the [[wikipedia:Gas_constant|gas constant]], <math>T</math> is the temperature, and <math>z</math> is the number of exchanged electrons per reaction.
<math>u(t)=v(t)-v^0</math> is the difference between the electrode potential and the reference potential.
This difference, to which we refer below as voltage, changes during the measurement of a [[wikipedia:Voltammetry|voltammogram]].

==Model==

The control volume method has been used for the spatial discretization of (1).
Together with the boundary conditions, the resulting system of ordinary differential equations is as follows,

:<math>
E\frac{d\vec{c}}{dt}+K(u(t))\vec{c}-A\vec{c}=B,\quad
y(t)=C\vec{c},\quad
\vec{c}(0)=\vec{c}_0 \neq 0,
</math>

where E and <math>K(u(t))</math> are system matrices, <math>K(u(t))</math> is a function of voltage that in turn depends on time.
The voltage appears in the system matrix due to the boundary conditions (2).
The vector <math>\vec{c} \in \mathbb{R}^n</math> is the vector of unknown concentrations, which includes both the <math>Ox</math> and <math>Red</math> species.
The vector <math>B</math> is the load vector, which arises as a consequence of the Dirichlet boundary conditions imposed at the bulk boundary of the electrolyte. The total current is computed as an integral (sum) over the electrode surface.
The matrix <math>K(u(t))</math> has the following form,

:<math>
K(u(t))=K_1(u(t))+K_2(u(t)),
</math>

where <math>K_i(u(t))=h_i D_i, \, i=1,2,</math> and <math>h_1=\exp(\beta u(t)), \, h_2=\exp(-\beta u(t))</math>.
The voltage <math>u(t)</math> is a function of <math>\sigma</math>,

:<math>
u(t)=\sigma t-1, \, \text{for } t \leq \frac{2}{ \sigma}, \quad
u(t)=-\sigma t+3, \, \text{for } \frac{2}{ \sigma} < t \leq \frac{4}{ \sigma},
</math>

where <math>\sigma</math> can take four different values, <math>\sigma=0.5, \, 0.05, \, 0.005, \, 0.0005</math>. The constant <math>\beta</math> is computed from the parameters <math>\alpha, \, z, \, F, \, R,</math> and <math>T,</math> leading to the value <math>\beta=21.243036728240824</math>.
Although the system is a time-varying system, it can be considered as a parametrized system with two parameters <math>h_1</math> and <math>h_2</math>.

==Data==

The data of the system matrices <math>E, \ D_1, \ D_2, \ A, \ B, C</math> as well as the initial state <math>\vec{c}_0=x_0</math> are in MatrixMarket format (http://math.nist.gov/MatrixMarket/), and can be downloaded here [[Media:SECM.TGZ|SECM.tgz]]. The quantity of interest is the current which is computed by <math>I(t)=C(5,:)\vec{c}</math> in MATLAB notation. The associated plot is called the [[wikipedia:Cyclic_voltammetry|cyclic voltammogram]] <ref name="feng06"/>, which is the plot of the current changing with the voltage <math>u(t)</math>.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
E\dot{c}(t) &=& (A_1 - h_1 A_2 - h_2 A_3)c(t) + Bu(t) \\
y(t) &=& Cc(t)
\end{array}
</math>

System dimensions:

<math>E \in \mathbb{R}^{16\,912 \times 16\,912}</math>,
<math>A_{1,2,3} \in \mathbb{R}^{16\,912 \times 16\,912}</math>,
<math>B \in \mathbb{R}^{16\,912 \times 1}</math>,
<math>C \in \mathbb{R}^{5 \times 16\,912}</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Scanning Electrochemical Microscopy'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Scanning_Electrochemical_Microscopy

@MISC{morwiki_secm,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Scanning Electrochemical Microscopy},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Scanning_Electrochemical_Microscopy}</nowiki>,
year = {2018}
}

* For the background on the benchmark:

@ARTICLE{morFenKRetal06,
author = {L. Feng, D. Koziol, E.B. Rudnyi, and J.G. Korvink},
title = {Parametric Model Reduction for Fast Simulation of Cyclic Voltammograms},
journal = {Sensor Letters},
volume = 4,
number = 2,
pages = {165--173},
year = 2006,
doi = {10.1166/sl.2006.021}
}

==References==

<references>

<ref name="mirkin01"> M.V. Mirkin, "[https://doi.org/10.1201/b11850-7 Chapter 5: Theory]", In: A.J. Bard and M.V. Mirkin, (eds.), Scanning Electrochemical Microscopy, CRC Press: 144--199, 2001.</ref>

<ref name="feng06"> L. Feng, D. Koziol, E.B. Rudnyi, and J.G. Korvink, "[https://doi.org/10.1166/sl.2006.021 Parametric Model Reduction for Fast Simulation of Cyclic Voltammograms]", Sensor Letters, 4(2): 165--173, 2006.</ref>

</references>

==Contact==

'' [[User:Feng|Lihong Feng]] ''

Anemometer

2023-05-02T13:52:01Z

Lund: /* Dimensions */ simplify matrix indexing

[[Category:benchmark]]
[[Category:parametric]]
[[Category:linear]]
[[Category:ODE]]
[[Category:time invariant]]
[[Category:Parametric]]
[[Category:affine parameter representation]]
[[Category:first differential order]]
[[Category:SISO]]

==Description==

<figure id="fig:plot1">
[[File:Model_Color.pdf|600px|thumb|right|<caption>Schematic 2D-Model-Anemometer</caption>]]
</figure>
<figure id="fig:plot2">

[[file:ContourPlot30.pdf|600px|thumb|right|<caption>Calculated temperature profile for the Anemometer function</caption>]]

</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)"/> (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 Fig. 1.
They are located on a membrane to minimize heat dissipation through the structure.
Without any flow, the heat dissipates symmetrically into 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 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)"/>:

:<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,
<math>v</math> is the fluid velocity, <math>T</math> is the temperature, and <math>\dot q</math> is the heat flow into the system caused by the heater.

The solid model has been generated and meshed in [[wikipedia:ANSYS|ANSYS]].
Triangular [http://www.ansys.stuba.sk/html/elem_55/chapter4/ES4-55.htm PLANE55] elements have been used for the finite element discretization.
The order of the system is <math>n = 29008</math>.

Example with one parameter:

The <math>n</math> dimensional [[wikipedia:Ordinary_Differential_Equation|ODE]] system has the following transfer function
:<math>
G(p) = C((sE - A_1- p(A_2 - A_1))^{-1}B)
</math>

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.
<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>.

Example with three parameters:

Here, all fluid properties are identified as parameters. Thus, we consider the following transfer function

:<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)
</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>.

==Origin==

* [http://www.imtek.uni-freiburg.de/professuren/simulation/ IMTEK Freiburg, Simulation group], Prof Dr Jan G. Korvink has taken on a position as Director of the Institute of Microstructure Technology (IMT) at the Karlsruhe Institute of Technology (KIT).

==Data==

Matrices are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format.
All matrices (for the one parameter system and for the three parameter case) can be found and uploaded in [[Media:Anemometer.tar.gz|Anemometer.tar.gz]].
The matrix name is used as an extension of the matrix file.
The system matrices have been extracted from ANSYS models by means of [http://simulation.uni-freiburg.de/downloads/mor4fem mor4fem].
For more information about computing the system matrices, the choice of the output, applying the permutation, please look into the [[media:Readme2.pdf|readme file]]. [[File: Readme2.pdf|thumb]]

Example with one parameter:

* <tt>.B</tt>: load vector
* <tt>.E</tt>: heat capacitance matrix
* <tt>.P</tt>: permutation matrix
* <tt>.A</tt>: stiffness matrices (2)

Example with three parameters:

* <tt>.B</tt>: load vector
* <tt>.E</tt>: heat capacitance matrices (2)
* <tt>.A</tt>: stiffness matrices (5)

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)"/>.

The output matrix <math>C \in \mathbb{R}^{1 \times 29008}</math> is a vector with non-zero elements <math>C_{173} = 1</math> and <math>C_{133} = -1</math>.

==Dimensions==

System structure (1 parameter):

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

System dimensions:

<math>E \in \mathbb{R}^{29008 \times 29008}</math>,
<math>A_{1,2} \in \mathbb{R}^{29008 \times 29008}</math>,
<math>B \in \mathbb{R}^{29008 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 29008}</math>.

System structure (3 parameter):

:<math>
\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 \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>E_{1,2} \in \mathbb{R}^{29008 \times 29008}</math>,
<math>A_{1,2,3,4,5} \in \mathbb{R}^{29008 \times 29008}</math>,
<math>B \in \mathbb{R}^{29008 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 29008}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_anemom,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Anemometer},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Anemometer}</nowiki>,
year = {2018}
}

* For the background on the benchmark:

==References==

a) About the '''Anemometer'''

<references group="a)">

<ref name="ernst01" group="a)">H. Ernst, "[http://www.freidok.uni-freiburg.de/volltexte/201/ High-Resolution Thermal Measurements in Fluids]," PhD thesis, University of Freiburg, Germany (2001).</ref>

<ref name="benner05" group="a)">P. Benner, V. Mehrmann and D. Sorensen, "[http://dx.doi.org/10.1007/3-540-27909-1 Dimension Reduction of Large-Scale Systems]", Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin/Heidelberg, Germany, 45, 2005.</ref>

<ref name="moosmann05" group="a)">C. Moosmann and A. Greiner, "[http://dx.doi.org/10.1007/3-540-27909-1_16 Convective Thermal Flow Problems]", Chapter 16 (pages 341--343) of 2.</ref>

</references>

b) MOR for non-parametrized '''Anemometer'''

<references group="b)">

<ref name="moosmann04" group="b)">C. Moosmann, E. B. Rudnyi, A. Greiner and J. G. Korvink, "[http://modelreduction.com/doc/papers/moosmann04THERMINIC.pdf Model Order Reduction for Linear Convective Thermal Flow]",
Proceedings of 10th International Workshops on THERMal INvestigations of ICs and Systems, THERMINIC2004, 29 Sept - 1 Oct, 2004, Sophia Antipolis, France.</ref>

</references>

c) MOR for parametrized '''Anemometer'''

<references group="c)">

<ref name="baur11" group="c)">U. Baur, P. Benner, A. Greiner, J. G. Korvink, J. Lienemann and C. Moosmann, "[http://dx.doi.org/10.1080/13873954.2011.547658 Parameter preserving model order reduction for MEMS applications]", MCMDS Mathematical and Computer Modeling of Dynamical Systems, 17(4):297--317, 2011.</ref>

<ref name="moosmann07" group="c)">C. Moosmann, "[http://www.freidok.uni-freiburg.de/volltexte/3971/ ParaMOR - Model Order Reduction for parameterized MEMS applications]", PhD thesis, University of Freiburg, Germany (2007).</ref>

<ref name="moosmann05b" group="c)">C. Moosmann, E. B. Rudnyi, A. Greiner, J. G. Korvink and M. Hornung, "[http://modelreduction.com/doc/papers/moosmann05MSM.pdf Parameter Preserving Model Order Reduction of a Flow Meter]", Technical Proceedings of the 2005 Nanotechnology
Conference and Trade Show, Nanotech 2005, May 8-12, 2005, Anaheim, California, USA, NSTINanotech
2005, vol. 3, p. 684-687.</ref>

<ref name="rudnyi06" group="c)">E. B. Rudnyi, C. Moosmann, A. Greiner, T. Bechtold, J. G. Korvink, "[http://modelreduction.com/doc/papers/rudnyi06mathmod.pdf Parameter Preserving Model Reduction for MEMS System-level Simulation and Design]", Proceedings of MATHMOD 2006, February 8 -
10, 2006, Vienna University of Technology, Austria.</ref>

</references>

==Contact==

[[User:Himpe]]

Thermal Model

2023-04-26T12:48:52Z

Lund: /* Dimensions */ update with .mat indices

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:first differential order]]
[[Category:time invariant]]
[[Category:MIMO]]
[[Category:Sparse]]

==Description: Boundary Condition Independent Thermal Model==
<figure id="fig1">[[File:ThermalModel.jpg|490px|thumb|right|<caption>A 2D-axisymmetrical model of the micro-thruster unit (not scaled). A heater is shown by a red spot.</caption>]]</figure>
A benchmark for the heat transfer problem with variable [[wikipedia:Heat_transfer_coefficient|film coefficients]] is presented.
It can be used to apply parametric model reduction algorithms to a linear first-order problem.

===Modeling===

An important requirements for a compact thermal model is that it should be independent from the boundary condition.
This means that a chip producer does not know conditions under which the chip will be used and hence the chip compact thermal model must allow an engineer to research on how the change in the environment influences the chip temperature.
The chip benchmarks representing boundary condition independent requirements are described in <ref name="lasance2001"/>.

Let us briefly describe the mathematical problem.
The thermal problem can be modeled by the heat transfer partial differential equation

:<math id="eq1">
\begin{align}
\nabla \cdot (\kappa(r)\nabla T(r,t)) + Q(r,t) - \rho(r)C_{p}(r)\frac{\partial T(r,t)}{\partial t} & = 0, & (1)
\end{align}
</math>

with <math>r</math> is the position, <math>t</math> is the time, <math>\kappa</math> is the thermal conductivity of the material, <math>C_{p}</math> is the specific heat capacity, <math>\rho</math> is the mass density, <math>Q</math> is the heat generation rate, and <math>T</math> is the unknown temperature distribution to be determined.
The heat exchange through device interfaces is usually modeled by convection boundary conditions

:<math id="eq2">
\begin{align}
q & = h_{i}(T - T_{bulk}), & (2)
\end{align}
</math>

where <math>q</math> is the heat flow through a given point, <math>h_{i}</math> is the film coefficient to describe the heat exchange for the <math>i</math>-th interface, <math>T</math> is the local temperature at this point, and <math>T_{bulk}</math> is the bulk temperature in the neighboring phase (in most cases <math>T_{bulk} = 0</math>).

After the discretization of equations [[#eq1|(1)]] and [[#eq2|(2)]] one obtains a system of ordinary differential equations as follows

:<math id="eq3">
\begin{align}
E \dot{x}(t) & = (A + \sum_{i} h_{i} A_{i})x(t) + B, & (3)
\end{align}
</math>

where <math>E</math> and <math>A</math> are the device system matrices, <math>A_i</math> is the diagonal matrix due to the discretization of equation [[#eq2|(2)]] for the <math>i</math>-th interface, <math>x(t)</math> is the vector with unknown temperatures.

In terms of the equation [[#eq3|(3)]] above, the engineering requirements read as follows.
A chip producer specifies the system matrices but the film coefficient, <math>h_i</math>, is controlled later on by another engineer.
As such, any reduced model to be useful should preserve <math>h_i</math> in the symbolic form.
This problem can be mathematically expressed as parametric model reduction<ref name="weile1999"/><ref name="gunupudi2003"/><ref name="daniel2004"/>.

Unfortunately, the benchmark from <ref name="lasance2001"/> is not available in the computer readable format.
For research purposes, we have modified a [[Micropyros_Thruster|Micropyros Thruster benchmark]] (see Fig. 1).
In the context of the present work, the model is as a generic example of a device with a single heat source when the generated heat dissipates through the device to the surroundings. The exchange between surrounding and the device is modeled by convection boundary conditions with different film coefficients at the top, <math>h_{top}</math>, bottom, <math>h_{bottom}</math>, and the side, <math>h_{side}</math>.
From this viewpoint, it is quite similar to a chip model used as a benchmark in <ref name="lasance2001"/>.
The goal of parametric model reduction in this case is to preserve <math>h_{top}</math>, <math>h_{bottom}</math>, and <math>h_{side}</math> in the reduced model in the symbolic form.

===Discretization===

We have used a 2D-axisymmetric microthruster model (T2DAL in [[Micropyros_Thruster|Micropyros Thruster]]).

The model has been made in [http://www.ansys.com/ ANSYS] and system matrices have been extracted by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem] <ref name="rudnyi2004"/>.
The benchmark contains a constant load vector.
The input function equal to one corresponds to the constant input power of <math>15 mW</math>.

The linear ordinary differential equations of first order are written as:

:<math>
\begin{align}
E \dot{x}(t) & = (A - h_{top} A_{top} - h_{bottom} A_{bottom} - h_{side} A_{side}) x(t) + B \\
y & = Cx(t)
\end{align}
</math>

where <math>E</math> and <math>A</math> are the symmetric sparse system matrices (heat capacity and heat conductivity matrix), <math>B</math> is the load vector, <math>C</math> is the output matrix, <math>A_{top}</math>, <math>A_{bottom}</math>, and <math>A_{side}</math> are the diagonal matrices from the discretization of the convection boundary conditions and <math>T</math> is the vector of unknown temperatures.

The numerical values of film coefficients, <math>h_{top}</math>, <math>h_{bottom}</math>, and <math>h_{side}</math> can be from <math>1</math> to <math>10^9</math>.
Typical important sets film coefficients can be found in <ref name="lasance2001"/>.
The allowable approximation error is <math>5\%</math> <ref name="lasance2001"/>.

The benchmark has been used in <ref name="feng2004"/><ref name="feng2005"/> where the problem is also described in more detail.

==Acknowledgements==

This work was partially funded by the DFG project '''MST-Compact (KO-1883/6)''', the Italian research council CNR together with the Italian province of Trento PAT, by the German Ministry of Research BMBF (SIMOD), and an operating grant of the University of Freiburg.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38865, see <ref name="rudnyi2005"/>.

==Data==

The matrices of this benchmark can be downloaded in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format:

* [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ThermalModel-dim1e3-T2DAL_BCI.tar.gz ThermalModel-dim1e3-T2DAL_BCI.tar.gz], 218.7 kB.

The matrix name is used as an extension of the matrix file. File <tt>T2DAL_BCI.C.names</tt> contains a list of output names written consecutively.

==Dimensions==

System structure:

:<math>
\begin{align}
E \dot{x}(t) & = (A_1 + h_2 A_2 + h_3 A_3 + h_4 A_4) x(t) + B \\
y(t) & = Cx(t)
\end{align}
</math>

System dimensions:

<math>E \in \mathbb{R}^{4257 \times 4257}</math>,
<math>A_1 \in \mathbb{R}^{4257 \times 4257}</math>,
<math>A_2 \in \mathbb{R}^{4257 \times 4257}</math>,
<math>A_3 \in \mathbb{R}^{4257 \times 4257}</math>,
<math>A_4 \in \mathbb{R}^{4257 \times 4257}</math>,
<math>B \in \mathbb{R}^{4257 \times 1}</math>,
<math>C \in \mathbb{R}^{7 \times 4257}</math>.

Parameter ranges:

<math>h_2 \in [1,10^9]</math>,
<math>h_3 \in [1,10^9]</math>,
<math>h_4 \in [1,10^9]</math>.

In the original Matrix Market data, <math>A_{top} = A_2, A_{bottom} = A_3, A_{side} = A_4</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection, '''Thermal Model'''. hosted at MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Thermal_Model

@MISC{morwiki_thermal,
author = <nowiki>{{Oberwolfach Benchmark Collection}}</nowiki>,
title = {Thermal Model},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Thermal_Model}</nowiki>,
year = 20XX
}

* For the background on the benchmark:

@TechReport{morFenRK04,
author = {Feng, L. and Rudnyi, E.~B. and Korvink, J.~G.},
title = {Parametric Model Reduction to Generate Boundary Condition Independent Compact Thermal Model},
institution = {IMTEK-Institute for Microsystem Technology},
type = {Technical report},
year = {2004},
url = <nowiki>{http://modelreduction.com/doc/papers/feng04THERMINIC.pdf}</nowiki>
}

==References==

<references>

<ref name="lasance2001">C.J.M. Lasance, "[https://doi.org/10.1109/6144.974943 Two benchmarks to facilitate the study of compact thermal modeling phenomena]", IEEE Transactions on Components and Packaging Technologies, 24: 559--565, 2001.</ref>

<ref name="weile1999">D.S. Weile, E. Michielssen, E. Grimme, K. Gallivan, "[https://doi.org/10.1016/S0893-9659(99)00063-4 A method for generating rational interpolant reduced order models of two-parameter linear systems]", Applied Mathematics Letters, 12: 93--102, 1999.</ref>

<ref name="gunupudi2003">P. K. Gunupudi, R. Khazaka, M. S. Nakhla, T. Smy, and D. Celo, "[https://doi.org/10.1109/TMTT.2003.820169 Passive parameterized time-domain macromodels for high-speed transmission-line networks]", IEEE Transactions on Microwave Theory and Techniques, 51: 2347--2354, 2003.</ref>

<ref name="daniel2004">L. Daniel, O.C. Siong, L.S. Chay, K.H. Lee, and J. White, "[https://doi.org/10.1109/TCAD.2004.826583 A Multiparameter Moment-Matching Model-Reduction Approach for Generating Geometrically Parameterized Interconnect Performance Models]", IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 23: 678--693, 2004.</ref>

<ref name="rudnyi2004">E.B. Rudnyi and J.G. Korvink, "[https://www.math.ucsd.edu/~helton/MTNSHISTORY/CONTENTS/2004LEUVEN/CDROM/papers/513.pdf Model Order Reduction of MEMS for Efficient Computer Aided Design and System Simulation]", MTNS2004, Sixteenth International Symposium on Mathematical Theory of Networks and Systems, Katholieke Universiteit Leuven, Belgium, July 5-9, 2004.</ref>

<ref name="feng2004">L. Feng, E.B. Rudnyi, J.G. Korvink, "[http://modelreduction.com/doc/papers/feng04THERMINIC.pdf Parametric Model Reduction to Generate Boundary Condition Independent Compact Thermal Model]", THERMINIC 2004, 10th International Workshop on Thermal Investigations of ICs and Systems, 29 September - 1 October 2004, Sophia Antipolis, Cote d'Azur, France.</ref>

<ref name="feng2005">L. Feng, E. B. Rudnyi, J. G. Korvink, "[https://doi.org/10.1109/TCAD.2005.852660 Preserving the film coefficient as a parameter in the compact thermal model for fast electro-thermal simulation]", IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 24(12): 1838--1847, 2005.</ref>

<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="rudnyi2005">E.B. Rudnyi, J.G. Korvink, [https://doi.org/10.1007/3-540-27909-1_17 Boundary Condition Independent Thermal Model], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 345--348, 2005.</ref>

</references>

==Contact==

'' [[User:Feng|Lihong Feng]] ''

Bone Model

2023-04-25T15:46:33Z

Lund: /* Data */ Correct typos

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

==Description: Trabecular Bone Micro-Finite Element Models==

<figure id="fig1">[[File:Bone1.jpg|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Bone2.jpg|490px|thumb|right|Figure 2]]</figure>

Three-dimensional serial reconstruction techniques allow us to develop very detailed micro-finite element (micro-FE) model of bones that can very accurately represent the porous bone micro-architecture.
Fig. 1 sketches the micro finite element analysis <ref name="rietbergen1995"/>.
Micro [[wikipedia:CT_scan|computed tomography]] (CT) is employed to make 3D high-resolution images (~50 microns) of a bone.
Then the 3D reconstruction is directly transformed into an equally shaped micro finite element model by simply converting all bone [[wikipedia:Voxel|voxels]] to equally sized 8-node brick elements.
This results in [[wikipedia:Finite_Element_Method|finite element]] (FE) models with a very large number of elements.
Such models can be used, for example to study differences in bone tissue loading between healthy and [[wikipedia:Osteoporosis|osteoporotic]] human bones during quasi static loading <ref name="rietbergen2003"/>.

There is increasing evidence, however, that bone responds in particular to dynamic loads <ref name="lanyon1984"/>.
It has been shown that the application of high-frequency, very low magnitude strains to a bone can prevent bone loss due to osteoporosis and can even result in increased bone strength in bones that are already osteoporotic.
In order to better understand this phenomenon, it is necessary to determine the strain as sensed by the bone cells due to this loading.
This would be possible with the micro-FE analysis, but then such an analysis need to be a dynamic one.

The present benchmark presents six bone models varying in dimension from about two-hundred-thousand to twelve millions equations with the goal to research on scalability of model reduction software.
Each model represents a second order system of the form

:<math>
\begin{array}{rcl}
M \ddot{x}(t) + K x(t) &=& B \\
y(t) &=& Cx(t)
\end{array}
</math>

where the matrices <math>M</math> and <math>K</math> are symmetric and positive definite.
The goal of model reduction is to speed up harmonic response analysis in the frequency range <math>1-100 Hz</math>.

The matrix properties are given in Table 1 below.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Bone micro-finite element models.''
|-
!
!BS01
!BS10
!B010
!B025
!B050
!B120
|-
|Number of Elements
|20098
|192539
|278259
|606253
|1378782
|3387547
|-
|Number of Nodes
|42508
|305066
|329001
|719987
|1644848
|3989996
|-
|Number of DoFs
|127224
|914898
|986703
|2159661
|4934244
|11969688
|-
|nnz in half M
|1182804
|9702186
|12437739
|27150810
|61866069
|151251738
|-
|nnz in half K
|3421188
|28191660
|36326514
|79292769
|180663963
|441785526
|-
|File
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e6-BS01.dat.gz BoneModel-dim1e6-BS01.dat.gz] (305.1 kB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-BS10.dat.gz BoneModel-dim1e7-BS10.dat.gz] (2.8 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B010.dat.gz BoneModel-dim1e7-B010.dat.gz] (3.9 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B025.dat.gz BoneModel-dim1e7-B025.dat.gz] (8.6 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B050.dat.gz BoneModel-dim1e7-B050.dat.gz] (19.6 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e8-B120.dat.gz BoneModel-dim1e8-B120.dat.gz] (48.5 MB)
|}

It should be stressed that the first two models have been obtained differently and they are much simpler to deal with than the last four.
The connectivity in the last four models is about four times higher.
This can be seen by comparing models <tt>BS10</tt> and <tt>B010</tt>.
Although models look similar by number of nonzeros in the system matrices, the model <tt>B010</tt> is much harder to solve:
The number of nonzero elements in the factor for model <tt>B010</tt> is about four times more than for <tt>BS10</tt>.

The method allows for the compact representation of the models, as the element mass and stiffness matrices are the same for all elements.
As a result, a file describing the node indices for each element is enough to assemble the global matrix.
Each node has three degrees of freedom (<math>UX</math>, <math>UY</math>, <math>UZ</math>) and it contributes three consecutive entries to the state vector.
The node numbering is natural from the first to the last.
The assembly procedure as a pseudo-code is presented below (indices start from one).
It is assumed that the last <math>300</math> degrees of freedom are fixed as zero Dirichlet boundary conditions.
For simplicity, the pseudo-code does not take into account that the matrix is symmetric.

==Origin==

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

==Data==

The data file for each model contains the number of elements, <tt>nel</tt>, and the number of nodes, <tt>nnod</tt>, in the first line and then <tt>nel</tt> number of lines with eight numbers for node indices in each line.

1) Read the element stiffness matrix elemK(24,24), 8 nodes * 3 degrees of freedom per node.
2) Read the number of elements, nel, and number of node, nnod, from the first line of the data file.
3) Number of degrees of freedom, ndof = nnod *3 - 300.
4) Allocate space for the sparse global matrix, matK(ndof, ndof).
5) Assembly:
Do (k = 1, nel)
Read eight node numbers from the k-th line, nodeindex(8);
Construct the index for degrees of freedom, dofindex(24):
Do (i = 1, 8)
Do (j = 1, 3)
dofindex((i - 1)*3 + j)= (nodeindex(i) - 1)*3 + j;
Use dofindex to assemble the element matrix elemK:
Do (i = 1, 24)
Do (j = 1, 24)
If (dofindex(i) < ndof AND dofindex(j) < ndof)
matK(dofindex(i), dofindex(j)) += elemK(i, j)

The input matrix contains a single column with <math>B(1) = 1</math>.
The output matrix takes the first three components of the state vector, that is, three displacements <math>UX</math>, <math>UY</math> and <math>UZ</math> for the first node.

The archive [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-assemble.tar.gz BoneModel-assemble.tar.gz] contains the element mass and stiffness matrices as well as the sample code in [[wikipedia:C++|C++]] to assemble the dynamic system.
The code can write the dynamic system in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format or can be used as a hook to transform the global matrices to an appropriate format.
The gzipped compressed data files for element assembly as described above can be downloaded from Table 1.

Model reduction for models <tt>BS010</tt> and <tt>BS10</tt> was performed in <ref name="rudnyi2005"/>. The benchmarking of the parallel [http://graal.ens-lyon.fr/MUMPS/ MUMPS] direct solver <ref name="amestoy2006"/> for the stiffness matrices is described in <ref name="rudnyi2006"/>.

Files (in .mat format) can be found for the three smaller systems ([https://sparse.tamu.edu/Oberwolfach/boneS01 BS01], [https://sparse.tamu.edu/Oberwolfach/boneS10 BS10], and [https://sparse.tamu.edu/Oberwolfach/bone010 B010]) on SuiteSparse.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
M \ddot{x}(t) + K x(t) &=& b \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>M \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}^{3 \times N}</math>.

System variants:

<tt>BS01</tt>: <math>N = 127224</math>,
<tt>BS10</tt>: <math>N = 914898</math>,
<tt>B010</tt>: <math>N = 986703</math>,
<tt>B025</tt>: <math>N = 2159661</math>,
<tt>B050</tt>: <math>N = 4934244</math>,
<tt>B120</tt>: <math>N = 11969688</math>.

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

@ARTICLE{RieWHetal95,
author = <nowiki>{B. van Rietbergen and H. Weinans and R. Huiskes and A. Odgaard}</nowiki>,
title = {A new method to determine trabecular bone elastic properties and loading using micromechanical finite-elements models},
journal = {Journal of Biomechanics},
volume = {28},
number = {1},
pages = {69--81},
year = {1995},
doi = {10.1016/0021-9290(95)80008-5}
}

==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="rietbergen1995">B. van Rietbergen, H. Weinans, R. Huiskes, A. Odgaard, [https://doi.org/10.1016/0021-9290(95)80008-5 A new method to determine trabecular bone elastic properties and loading using micromechanical finite-elements models] Journal of Biomechanics, 28(1): 69--81, 1995.</ref>

<ref name="rietbergen2003">B. van Rietbergen, R. Huiskes, F. Eckstein, P. Rueegsegger, [https://doi.org/10.1359/jbmr.2003.18.10.1781 Trabecular Bone Tissue Strains in the Healthy and Osteoporotic Human Femur], Journal of Bone and Mineral Research, 18(10): 1781--1787, 2003.</ref>

<ref name="lanyon1984">L.E. Lanyon, C.T. Rubin, [https://doi.org/10.1016/0021-9290(84)90003-4 Static versus dynamic loads as an influence on bone remodelling], Journal of Biomechanics, 17: 897--906, 1984.</ref>

<ref name="rudnyi2005">E.B. Rudnyi, B. van Rietbergen, J.G. Korvink. [http://repository.tue.nl/b3454bd9-f190-4c33-8dca-22c50b08edec Efficient Harmonic Simulation of a Trabecular Bone Finite Element Model by means of Model Reduction]. 12th Workshop "The Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields", Proceedings of the 12th FEM Workshop: 61--68, 2005</ref>

<ref name="amestoy2006">P.R. Amestoy, A. Guermouche and J.-Y. L'Excellent, S. Pralet, [https://doi.org/10.1016/j.parco.2005.07.004 Hybrid scheduling for the parallel solution of linear systems]. Parallel Computing, 32(2): 136--156, 2006.</ref>

<ref name="rudnyi2006">E.B. Rudnyi, B. van Rietbergen, J. G. Korvink. [http://modelreduction.com/doc/papers/rudnyi06tam.pdf Model Reduction for High Dimensional Micro-FE Models]. TAM'06, The Third HPC-Europa Transnational Access Meeting, Barcelona, 2006.</ref>

</references>

Bone Model

2023-04-25T15:46:10Z

Lund: Added links to SuiteSparse

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

==Description: Trabecular Bone Micro-Finite Element Models==

<figure id="fig1">[[File:Bone1.jpg|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Bone2.jpg|490px|thumb|right|Figure 2]]</figure>

Three-dimensional serial reconstruction techniques allow us to develop very detailed micro-finite element (micro-FE) model of bones that can very accurately represent the porous bone micro-architecture.
Fig. 1 sketches the micro finite element analysis <ref name="rietbergen1995"/>.
Micro [[wikipedia:CT_scan|computed tomography]] (CT) is employed to make 3D high-resolution images (~50 microns) of a bone.
Then the 3D reconstruction is directly transformed into an equally shaped micro finite element model by simply converting all bone [[wikipedia:Voxel|voxels]] to equally sized 8-node brick elements.
This results in [[wikipedia:Finite_Element_Method|finite element]] (FE) models with a very large number of elements.
Such models can be used, for example to study differences in bone tissue loading between healthy and [[wikipedia:Osteoporosis|osteoporotic]] human bones during quasi static loading <ref name="rietbergen2003"/>.

There is increasing evidence, however, that bone responds in particular to dynamic loads <ref name="lanyon1984"/>.
It has been shown that the application of high-frequency, very low magnitude strains to a bone can prevent bone loss due to osteoporosis and can even result in increased bone strength in bones that are already osteoporotic.
In order to better understand this phenomenon, it is necessary to determine the strain as sensed by the bone cells due to this loading.
This would be possible with the micro-FE analysis, but then such an analysis need to be a dynamic one.

The present benchmark presents six bone models varying in dimension from about two-hundred-thousand to twelve millions equations with the goal to research on scalability of model reduction software.
Each model represents a second order system of the form

:<math>
\begin{array}{rcl}
M \ddot{x}(t) + K x(t) &=& B \\
y(t) &=& Cx(t)
\end{array}
</math>

where the matrices <math>M</math> and <math>K</math> are symmetric and positive definite.
The goal of model reduction is to speed up harmonic response analysis in the frequency range <math>1-100 Hz</math>.

The matrix properties are given in Table 1 below.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Bone micro-finite element models.''
|-
!
!BS01
!BS10
!B010
!B025
!B050
!B120
|-
|Number of Elements
|20098
|192539
|278259
|606253
|1378782
|3387547
|-
|Number of Nodes
|42508
|305066
|329001
|719987
|1644848
|3989996
|-
|Number of DoFs
|127224
|914898
|986703
|2159661
|4934244
|11969688
|-
|nnz in half M
|1182804
|9702186
|12437739
|27150810
|61866069
|151251738
|-
|nnz in half K
|3421188
|28191660
|36326514
|79292769
|180663963
|441785526
|-
|File
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e6-BS01.dat.gz BoneModel-dim1e6-BS01.dat.gz] (305.1 kB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-BS10.dat.gz BoneModel-dim1e7-BS10.dat.gz] (2.8 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B010.dat.gz BoneModel-dim1e7-B010.dat.gz] (3.9 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B025.dat.gz BoneModel-dim1e7-B025.dat.gz] (8.6 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B050.dat.gz BoneModel-dim1e7-B050.dat.gz] (19.6 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e8-B120.dat.gz BoneModel-dim1e8-B120.dat.gz] (48.5 MB)
|}

It should be stressed that the first two models have been obtained differently and they are much simpler to deal with than the last four.
The connectivity in the last four models is about four times higher.
This can be seen by comparing models <tt>BS10</tt> and <tt>B010</tt>.
Although models look similar by number of nonzeros in the system matrices, the model <tt>B010</tt> is much harder to solve:
The number of nonzero elements in the factor for model <tt>B010</tt> is about four times more than for <tt>BS10</tt>.

The method allows for the compact representation of the models, as the element mass and stiffness matrices are the same for all elements.
As a result, a file describing the node indices for each element is enough to assemble the global matrix.
Each node has three degrees of freedom (<math>UX</math>, <math>UY</math>, <math>UZ</math>) and it contributes three consecutive entries to the state vector.
The node numbering is natural from the first to the last.
The assembly procedure as a pseudo-code is presented below (indices start from one).
It is assumed that the last <math>300</math> degrees of freedom are fixed as zero Dirichlet boundary conditions.
For simplicity, the pseudo-code does not take into account that the matrix is symmetric.

==Origin==

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

==Data==

The data file for each model contains the number of elements, <tt>nel</tt>, and the number of nodes, <tt>nnod</tt>, in the first line and then <tt>nel</tt> number of lines with eight numbers for node indices in each line.

1) Read the element stiffness matrix elemK(24,24), 8 nodes * 3 degrees of freedom per node.
2) Read the number of elements, nel, and number of node, nnod, from the first line of the data file.
3) Number of degrees of freedom, ndof = nnod *3 - 300.
4) Allocate space for the sparse global matrix, matK(ndof, ndof).
5) Assembly:
Do (k = 1, nel)
Read eight node numbers from the k-th line, nodeindex(8);
Construct the index for degrees of freedom, dofindex(24):
Do (i = 1, 8)
Do (j = 1, 3)
dofindex((i - 1)*3 + j)= (nodeindex(i) - 1)*3 + j;
Use dofindex to assemble the element matrix elemK:
Do (i = 1, 24)
Do (j = 1, 24)
If (dofindex(i) < ndof AND dofindex(j) < ndof)
matK(dofindex(i), dofindex(j)) += elemK(i, j)

The input matrix contains a single column with <math>B(1) = 1</math>.
The output matrix takes the first three components of the state vector, that is, three displacements <math>UX</math>, <math>UY</math> and <math>UZ</math> for the first node.

The archive [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-assemble.tar.gz BoneModel-assemble.tar.gz] contains the element mass and stiffness matrices as well as the sample code in [[wikipedia:C++|C++]] to assemble the dynamic system.
The code can write the dynamic system in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format or can be used as a hook to transform the global matrices to an appropriate format.
The gzipped compressed data files for element assembly as described above can be downloaded from Table 1.

Model reduction for models <tt>BS010</tt> and <tt>BS10</tt> was performed in <ref name="rudnyi2005"/>. The benchmarking of the parallel [http://graal.ens-lyon.fr/MUMPS/ MUMPS] direct solver <ref name="amestoy2006"/> for the stiffness matrices is described in <ref name="rudnyi2006"/>.

Files (in .mat) format can be found for the three smaller systems ([https://sparse.tamu.edu/Oberwolfach/boneS01 BS01], [https://sparse.tamu.edu/Oberwolfach/boneS10 BS10], and [https://sparse.tamu.edu/Oberwolfach/bone010 B010]) on SuiteSparse

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
M \ddot{x}(t) + K x(t) &=& b \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>M \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}^{3 \times N}</math>.

System variants:

<tt>BS01</tt>: <math>N = 127224</math>,
<tt>BS10</tt>: <math>N = 914898</math>,
<tt>B010</tt>: <math>N = 986703</math>,
<tt>B025</tt>: <math>N = 2159661</math>,
<tt>B050</tt>: <math>N = 4934244</math>,
<tt>B120</tt>: <math>N = 11969688</math>.

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

@ARTICLE{RieWHetal95,
author = <nowiki>{B. van Rietbergen and H. Weinans and R. Huiskes and A. Odgaard}</nowiki>,
title = {A new method to determine trabecular bone elastic properties and loading using micromechanical finite-elements models},
journal = {Journal of Biomechanics},
volume = {28},
number = {1},
pages = {69--81},
year = {1995},
doi = {10.1016/0021-9290(95)80008-5}
}

==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="rietbergen1995">B. van Rietbergen, H. Weinans, R. Huiskes, A. Odgaard, [https://doi.org/10.1016/0021-9290(95)80008-5 A new method to determine trabecular bone elastic properties and loading using micromechanical finite-elements models] Journal of Biomechanics, 28(1): 69--81, 1995.</ref>

<ref name="rietbergen2003">B. van Rietbergen, R. Huiskes, F. Eckstein, P. Rueegsegger, [https://doi.org/10.1359/jbmr.2003.18.10.1781 Trabecular Bone Tissue Strains in the Healthy and Osteoporotic Human Femur], Journal of Bone and Mineral Research, 18(10): 1781--1787, 2003.</ref>

<ref name="lanyon1984">L.E. Lanyon, C.T. Rubin, [https://doi.org/10.1016/0021-9290(84)90003-4 Static versus dynamic loads as an influence on bone remodelling], Journal of Biomechanics, 17: 897--906, 1984.</ref>

<ref name="rudnyi2005">E.B. Rudnyi, B. van Rietbergen, J.G. Korvink. [http://repository.tue.nl/b3454bd9-f190-4c33-8dca-22c50b08edec Efficient Harmonic Simulation of a Trabecular Bone Finite Element Model by means of Model Reduction]. 12th Workshop "The Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields", Proceedings of the 12th FEM Workshop: 61--68, 2005</ref>

<ref name="amestoy2006">P.R. Amestoy, A. Guermouche and J.-Y. L'Excellent, S. Pralet, [https://doi.org/10.1016/j.parco.2005.07.004 Hybrid scheduling for the parallel solution of linear systems]. Parallel Computing, 32(2): 136--156, 2006.</ref>

<ref name="rudnyi2006">E.B. Rudnyi, B. van Rietbergen, J. G. Korvink. [http://modelreduction.com/doc/papers/rudnyi06tam.pdf Model Reduction for High Dimensional Micro-FE Models]. TAM'06, The Third HPC-Europa Transnational Access Meeting, Barcelona, 2006.</ref>

</references>

Nonlinear Heat Transfer

2023-04-24T16:43:51Z

Lund: /* Dimensions */ add F matrix

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

==Description==

The simulation of heat transport for a single device is easily tackled by current computational resources, even for a complex, finely structured geometry;
however, the calculation of a multi-scale system consisting of a large number of those devices, e.g., assembled printed circuit boards, is still a challenge.
A further problem is the large change in heat conductivity of many semiconductor materials with temperature.
We model the heat transfer along a 1D beam that has a nonlinear heat capacity which is represented by a polynomial of arbitrary degree as a function of the temperature state.
For accurate modelling of the temperature distribution, the resulting model requires many state variables to be described adequately.
The resulting complexity, i.e., number of first order differential equations and nonlinear parts, is such that a simplification or model reduction is needed in order to perform a simulation in an acceptable amount of time for the applications at hand.
Thus the need for model order reduction emerges.

===Model description===

We model the heat transfer along a 1D beam with length <math>L</math>, cross sectional area <math>A</math> and nonlinear heat conductivity represented by a polynomial in temperature <math>T(x,t)</math> of arbitrary degree <math>n</math>:

<math>
\kappa(T) = a_0 + a_1 T + a_2 T^2 + \dots
</math>

The output of the model is the temperature <math>T(x,t)</math>, the degrees of freedom are the temperature from left to right.
The right end of the beam (at <math>x=L</math>) is fixed at ambient temperature <math>0</math>;
this node does not occur in the model any more.
The model features two inputs: The first one is a time-dependent uniform heat flux <math>f</math> [W/m2] flowing in from the left end (at <math>x=0</math>).
The second one is a time dependent heat source <math>Q</math> [W/m3] in the beam volume, e.g. from an electric current.

===Benchmark examples===

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 <tt>DSIF</tt> format, which allows for nonlinear terms.

Three ready-made examples are available (all files are <tt>gzip</tt> compressed <tt>DSIF</tt> files, Units: SI):

====Linear example (heat conductivity not temperature dependent)====

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Specification of <tt>NonlinearHeatCond-n15-linear.zip</tt>.''
|-
|Property
|Symbol
|Unit
|Value
|-
|Number of nodes
|
|
|<math>15</math>
|-
|Beam length
|<math>l</math>
|[m]
|<math>0.1</math>
|-
|Cross-sectional area
|<math>A</math>
|[m2]
|<math>10^{-4}</math>
|-
|Material density
|<math>\rho</math>
|[kg/m3]
|<math>3970</math>
|-
|Heat capacity
|<math>C_p</math>
|[J/kg K]
|<math>766</math>
|-
|Heat conductivity
|<math>a_0</math>
|[W/m K]
|<math>36</math>
|}

====Nonlinear examples (heat conductivity temperature dependent)====

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Specification of <tt>NonlinearHeatCond-n15-nonlinear.zip</tt>.''
|-
|Property
|Symbol
|Unit
|Value
|-
|Number of nodes
|
|
|<math>15</math>
|-
|Beam length
|<math>l</math>
|[m]
|<math>0.1</math>
|-
|Cross-sectional area
|<math>A</math>
|[m2]
|<math>10^{-4}</math>
|-
|Material density
|<math>\rho</math>
|[kg/m3]
|<math>3970</math>
|-
|Heat capacity
|<math>C_p</math>
|[J/kg K]
|<math>766</math>
|-
|Heat conductivity
|<math>a_0</math>
|[W/m K]
|<math>36</math>
|-
|Heat conductivity
|<math>a1</math>
|[W/m K2]
|<math>-0.1116</math>
|-
|Heat conductivity
|<math>a_2</math>
|[W/m K3]
|<math>0.00017298</math>
|-
|Heat conductivity
|<math>a_3</math>
|[W/m K4]
|<math>-1.78746 \cdot 10^{-7}</math>
|-
|Heat conductivity
|<math>a_4</math>
|[W/m K5]
|<math>1.3852815 \cdot 10^{-10}</math>
|}

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Specification of <tt>NonlinearHeatCond-n410-nonlinear.zip</tt>.''
|-
|Property
|Symbol
|Unit
|Value
|-
|Number of nodes
|
|
|<math>410</math>
|-
|Beam length
|<math>l</math>
|[m]
|<math>0.1</math>
|-
|Cross-sectional area
|<math>A</math>
|[m2]
|<math>10^{-4}</math>
|-
|Material density
|<math>\rho</math>
|[kg/m3]
|<math>3970</math>
|-
|Heat capacity
|<math>C_p</math>
|[J/kg K]
|<math>766</math>
|-
|Heat conductivity
|<math>a_0</math>
|[W/m K]
|<math>36</math>
|-
|Heat conductivity
|<math>a_1</math>
|[W/m K2]
|<math>-0.1116</math>
|-
|Heat conductivity
|<math>a_2</math>
|[W/m K3]
|<math>0.00017298</math>
|-
|Heat conductivity
|<math>a_3</math>
|[W/m K4]
|<math>-1.78746 \cdot 10^{-7}</math>
|-
|Heat conductivity
|<math>a_4</math>
|[W/m K5]
|<math>1.3852815 \cdot 10^{-10}</math>
|}

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

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

The two outputs are on the left end and in the middle of the beam.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38883, see <ref name="lienemann05"/>.

==Data==

Details of the implementation are available in a separate report<ref name="lienemann04"/>.
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.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/NonlinearHeatTransfer-dim1e1-n15-linear.zip NonlinearHeatTransfer-dim1e1-n15-linear.zip]
|1 kB
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/NonlinearHeatTransfer-dim1e1-n15-nonlinear.zip NonlinearHeatTransfer-dim1e1-n15-nonlinear.zip]
|1.2 kB
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/NonlinearHeatTransfer-dim1e2-n410-nonlinear.zip NonlinearHeatTransfer-dim1e2-n410-nonlinear.zip]
|18.8 kB
|}

==Dimensions==

System structure:

:<math>
\begin{align}
E\dot{x}(t) &= Ax(t) + Bu(t) + F f(x(t)) \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times 2}</math>,
<math>C \in \mathbb{R}^{2 \times N}</math>,
<math>E \in \mathbb{R}^{N \times N}</math>,
<math>F \in \mathbb{R}^{N \times N}</math>,
<math>f : \mathbb{R}^N \to \mathbb{R}^N</math>.

System variants:

<tt>n15-linear</tt>: <math>N = 15</math>,
<tt>n15-nonlinear</tt>: <math>N = 15</math>,
<tt>n410-nonlinear</tt>: <math>N = 410</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Nonlinear Heat Transfer'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Nonlinear_Heat_Transfer

@MISC{morwiki_nheat,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Nonlinear Heat Transfer},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Nonlinear_Heat_Transfer}</nowiki>,
year = {20XX}
}

* For the background on the benchmark:

@INPROCEEDINGS{LieYK04,
author = <nowiki>{J. Lienemann, A. Yousefi, J.G. Korvink}</nowiki>,
title = {Nonlinear heat transfer modelling},
booktile = {12th Mediterranean Conference on Control and Automation},
year = {2004}
}

==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="lienemann05"> J. Lienemann, A. Yousefi, J.G. Korvink, [https://doi.org/10.1007/3-540-27909-1_13 Nonlinear Heat Transfer Modeling], In: Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 327--331, 2005.</ref>

<ref name="lienemann04"> J. Lienemann, A. Yousefi, J.G. Korvink, [https://www.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/NonlinearHeatTransfer.pdf Nonlinear heat transfer modelling], 12th Mediterranean Conference on Control and Automation, 2004.</ref>

</references>

Bone Model

2023-04-24T15:03:29Z

Lund: add s to BoneModel-assemble download link

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

==Description: Trabecular Bone Micro-Finite Element Models==

<figure id="fig1">[[File:Bone1.jpg|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Bone2.jpg|490px|thumb|right|Figure 2]]</figure>

Three-dimensional serial reconstruction techniques allow us to develop very detailed micro-finite element (micro-FE) model of bones that can very accurately represent the porous bone micro-architecture.
Fig. 1 sketches the micro finite element analysis <ref name="rietbergen1995"/>.
Micro [[wikipedia:CT_scan|computed tomography]] (CT) is employed to make 3D high-resolution images (~50 microns) of a bone.
Then the 3D reconstruction is directly transformed into an equally shaped micro finite element model by simply converting all bone [[wikipedia:Voxel|voxels]] to equally sized 8-node brick elements.
This results in [[wikipedia:Finite_Element_Method|finite element]] (FE) models with a very large number of elements.
Such models can be used, for example to study differences in bone tissue loading between healthy and [[wikipedia:Osteoporosis|osteoporotic]] human bones during quasi static loading <ref name="rietbergen2003"/>.

There is increasing evidence, however, that bone responds in particular to dynamic loads <ref name="lanyon1984"/>.
It has been shown that the application of high-frequency, very low magnitude strains to a bone can prevent bone loss due to osteoporosis and can even result in increased bone strength in bones that are already osteoporotic.
In order to better understand this phenomenon, it is necessary to determine the strain as sensed by the bone cells due to this loading.
This would be possible with the micro-FE analysis, but then such an analysis need to be a dynamic one.

The present benchmark presents six bone models varying in dimension from about two-hundred-thousand to twelve millions equations with the goal to research on scalability of model reduction software.
Each model represents a second order system of the form

:<math>
\begin{array}{rcl}
M \ddot{x}(t) + K x(t) &=& B \\
y(t) &=& Cx(t)
\end{array}
</math>

where the matrices <math>M</math> and <math>K</math> are symmetric and positive definite.
The goal of model reduction is to speed up harmonic response analysis in the frequency range <math>1-100 Hz</math>.

The matrix properties are given in Table 1 below.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Bone micro-finite element models.''
|-
!
!BS01
!BS10
!B010
!B025
!B050
!B120
|-
|Number of Elements
|20098
|192539
|278259
|606253
|1378782
|3387547
|-
|Number of Nodes
|42508
|305066
|329001
|719987
|1644848
|3989996
|-
|Number of DoFs
|127224
|914898
|986703
|2159661
|4934244
|11969688
|-
|nnz in half M
|1182804
|9702186
|12437739
|27150810
|61866069
|151251738
|-
|nnz in half K
|3421188
|28191660
|36326514
|79292769
|180663963
|441785526
|-
|File
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e6-BS01.dat.gz BoneModel-dim1e6-BS01.dat.gz] (305.1 kB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-BS10.dat.gz BoneModel-dim1e7-BS10.dat.gz] (2.8 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B010.dat.gz BoneModel-dim1e7-B010.dat.gz] (3.9 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B025.dat.gz BoneModel-dim1e7-B025.dat.gz] (8.6 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B050.dat.gz BoneModel-dim1e7-B050.dat.gz] (19.6 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e8-B120.dat.gz BoneModel-dim1e8-B120.dat.gz] (48.5 MB)
|}

It should be stressed that the first two models have been obtained differently and they are much simpler to deal with than the last four.
The connectivity in the last four models is about four times higher.
This can be seen by comparing models <tt>BS10</tt> and <tt>B010</tt>.
Although models look similar by number of nonzeros in the system matrices, the model <tt>B010</tt> is much harder to solve:
The number of nonzero elements in the factor for model <tt>B010</tt> is about four times more than for <tt>BS10</tt>.

The method allows for the compact representation of the models, as the element mass and stiffness matrices are the same for all elements.
As a result, a file describing the node indices for each element is enough to assemble the global matrix.
Each node has three degrees of freedom (<math>UX</math>, <math>UY</math>, <math>UZ</math>) and it contributes three consecutive entries to the state vector.
The node numbering is natural from the first to the last.
The assembly procedure as a pseudo-code is presented below (indices start from one).
It is assumed that the last <math>300</math> degrees of freedom are fixed as zero Dirichlet boundary conditions.
For simplicity, the pseudo-code does not take into account that the matrix is symmetric.

==Origin==

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

==Data==

The data file for each model contains the number of elements, <tt>nel</tt>, and the number of nodes, <tt>nnod</tt>, in the first line and then <tt>nel</tt> number of lines with eight numbers for node indices in each line.

1) Read the element stiffness matrix elemK(24,24), 8 nodes * 3 degrees of freedom per node.
2) Read the number of elements, nel, and number of node, nnod, from the first line of the data file.
3) Number of degrees of freedom, ndof = nnod *3 - 300.
4) Allocate space for the sparse global matrix, matK(ndof, ndof).
5) Assembly:
Do (k = 1, nel)
Read eight node numbers from the k-th line, nodeindex(8);
Construct the index for degrees of freedom, dofindex(24):
Do (i = 1, 8)
Do (j = 1, 3)
dofindex((i - 1)*3 + j)= (nodeindex(i) - 1)*3 + j;
Use dofindex to assemble the element matrix elemK:
Do (i = 1, 24)
Do (j = 1, 24)
If (dofindex(i) < ndof AND dofindex(j) < ndof)
matK(dofindex(i), dofindex(j)) += elemK(i, j)

The input matrix contains a single column with <math>B(1) = 1</math>.
The output matrix takes the first three components of the state vector, that is, three displacements <math>UX</math>, <math>UY</math> and <math>UZ</math> for the first node.

The archive [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-assemble.tar.gz BoneModel-assemble.tar.gz] contains the element mass and stiffness matrices as well as the sample code in [[wikipedia:C++|C++]] to assemble the dynamic system.
The code can write the dynamic system in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format or can be used as a hook to transform the global matrices to an appropriate format.
The gzipped compressed data files for element assembly as described above can be downloaded from Table 1.

Model reduction for models <tt>BS010</tt> and <tt>BS10</tt> was performed in <ref name="rudnyi2005"/>. The benchmarking of the parallel [http://graal.ens-lyon.fr/MUMPS/ MUMPS] direct solver <ref name="amestoy2006"/> for the stiffness matrices is described in <ref name="rudnyi2006"/>.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
M \ddot{x}(t) + K x(t) &=& b \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>M \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}^{3 \times N}</math>.

System variants:

<tt>BS01</tt>: <math>N = 127224</math>,
<tt>BS10</tt>: <math>N = 914898</math>,
<tt>B010</tt>: <math>N = 986703</math>,
<tt>B025</tt>: <math>N = 2159661</math>,
<tt>B050</tt>: <math>N = 4934244</math>,
<tt>B120</tt>: <math>N = 11969688</math>.

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

@ARTICLE{RieWHetal95,
author = <nowiki>{B. van Rietbergen and H. Weinans and R. Huiskes and A. Odgaard}</nowiki>,
title = {A new method to determine trabecular bone elastic properties and loading using micromechanical finite-elements models},
journal = {Journal of Biomechanics},
volume = {28},
number = {1},
pages = {69--81},
year = {1995},
doi = {10.1016/0021-9290(95)80008-5}
}

==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="rietbergen1995">B. van Rietbergen, H. Weinans, R. Huiskes, A. Odgaard, [https://doi.org/10.1016/0021-9290(95)80008-5 A new method to determine trabecular bone elastic properties and loading using micromechanical finite-elements models] Journal of Biomechanics, 28(1): 69--81, 1995.</ref>

<ref name="rietbergen2003">B. van Rietbergen, R. Huiskes, F. Eckstein, P. Rueegsegger, [https://doi.org/10.1359/jbmr.2003.18.10.1781 Trabecular Bone Tissue Strains in the Healthy and Osteoporotic Human Femur], Journal of Bone and Mineral Research, 18(10): 1781--1787, 2003.</ref>

<ref name="lanyon1984">L.E. Lanyon, C.T. Rubin, [https://doi.org/10.1016/0021-9290(84)90003-4 Static versus dynamic loads as an influence on bone remodelling], Journal of Biomechanics, 17: 897--906, 1984.</ref>

<ref name="rudnyi2005">E.B. Rudnyi, B. van Rietbergen, J.G. Korvink. [http://repository.tue.nl/b3454bd9-f190-4c33-8dca-22c50b08edec Efficient Harmonic Simulation of a Trabecular Bone Finite Element Model by means of Model Reduction]. 12th Workshop "The Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields", Proceedings of the 12th FEM Workshop: 61--68, 2005</ref>

<ref name="amestoy2006">P.R. Amestoy, A. Guermouche and J.-Y. L'Excellent, S. Pralet, [https://doi.org/10.1016/j.parco.2005.07.004 Hybrid scheduling for the parallel solution of linear systems]. Parallel Computing, 32(2): 136--156, 2006.</ref>

<ref name="rudnyi2006">E.B. Rudnyi, B. van Rietbergen, J. G. Korvink. [http://modelreduction.com/doc/papers/rudnyi06tam.pdf Model Reduction for High Dimensional Micro-FE Models]. TAM'06, The Third HPC-Europa Transnational Access Meeting, Barcelona, 2006.</ref>

</references>

Bone Model

2023-04-24T15:02:23Z

Lund: correct link to BS10

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

==Description: Trabecular Bone Micro-Finite Element Models==

<figure id="fig1">[[File:Bone1.jpg|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Bone2.jpg|490px|thumb|right|Figure 2]]</figure>

Three-dimensional serial reconstruction techniques allow us to develop very detailed micro-finite element (micro-FE) model of bones that can very accurately represent the porous bone micro-architecture.
Fig. 1 sketches the micro finite element analysis <ref name="rietbergen1995"/>.
Micro [[wikipedia:CT_scan|computed tomography]] (CT) is employed to make 3D high-resolution images (~50 microns) of a bone.
Then the 3D reconstruction is directly transformed into an equally shaped micro finite element model by simply converting all bone [[wikipedia:Voxel|voxels]] to equally sized 8-node brick elements.
This results in [[wikipedia:Finite_Element_Method|finite element]] (FE) models with a very large number of elements.
Such models can be used, for example to study differences in bone tissue loading between healthy and [[wikipedia:Osteoporosis|osteoporotic]] human bones during quasi static loading <ref name="rietbergen2003"/>.

There is increasing evidence, however, that bone responds in particular to dynamic loads <ref name="lanyon1984"/>.
It has been shown that the application of high-frequency, very low magnitude strains to a bone can prevent bone loss due to osteoporosis and can even result in increased bone strength in bones that are already osteoporotic.
In order to better understand this phenomenon, it is necessary to determine the strain as sensed by the bone cells due to this loading.
This would be possible with the micro-FE analysis, but then such an analysis need to be a dynamic one.

The present benchmark presents six bone models varying in dimension from about two-hundred-thousand to twelve millions equations with the goal to research on scalability of model reduction software.
Each model represents a second order system of the form

:<math>
\begin{array}{rcl}
M \ddot{x}(t) + K x(t) &=& B \\
y(t) &=& Cx(t)
\end{array}
</math>

where the matrices <math>M</math> and <math>K</math> are symmetric and positive definite.
The goal of model reduction is to speed up harmonic response analysis in the frequency range <math>1-100 Hz</math>.

The matrix properties are given in Table 1 below.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Bone micro-finite element models.''
|-
!
!BS01
!BS10
!B010
!B025
!B050
!B120
|-
|Number of Elements
|20098
|192539
|278259
|606253
|1378782
|3387547
|-
|Number of Nodes
|42508
|305066
|329001
|719987
|1644848
|3989996
|-
|Number of DoFs
|127224
|914898
|986703
|2159661
|4934244
|11969688
|-
|nnz in half M
|1182804
|9702186
|12437739
|27150810
|61866069
|151251738
|-
|nnz in half K
|3421188
|28191660
|36326514
|79292769
|180663963
|441785526
|-
|File
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e6-BS01.dat.gz BoneModel-dim1e6-BS01.dat.gz] (305.1 kB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-BS10.dat.gz BoneModel-dim1e7-BS10.dat.gz] (2.8 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B010.dat.gz BoneModel-dim1e7-B010.dat.gz] (3.9 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B025.dat.gz BoneModel-dim1e7-B025.dat.gz] (8.6 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B050.dat.gz BoneModel-dim1e7-B050.dat.gz] (19.6 MB)
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e8-B120.dat.gz BoneModel-dim1e8-B120.dat.gz] (48.5 MB)
|}

It should be stressed that the first two models have been obtained differently and they are much simpler to deal with than the last four.
The connectivity in the last four models is about four times higher.
This can be seen by comparing models <tt>BS10</tt> and <tt>B010</tt>.
Although models look similar by number of nonzeros in the system matrices, the model <tt>B010</tt> is much harder to solve:
The number of nonzero elements in the factor for model <tt>B010</tt> is about four times more than for <tt>BS10</tt>.

The method allows for the compact representation of the models, as the element mass and stiffness matrices are the same for all elements.
As a result, a file describing the node indices for each element is enough to assemble the global matrix.
Each node has three degrees of freedom (<math>UX</math>, <math>UY</math>, <math>UZ</math>) and it contributes three consecutive entries to the state vector.
The node numbering is natural from the first to the last.
The assembly procedure as a pseudo-code is presented below (indices start from one).
It is assumed that the last <math>300</math> degrees of freedom are fixed as zero Dirichlet boundary conditions.
For simplicity, the pseudo-code does not take into account that the matrix is symmetric.

==Origin==

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

==Data==

The data file for each model contains the number of elements, <tt>nel</tt>, and the number of nodes, <tt>nnod</tt>, in the first line and then <tt>nel</tt> number of lines with eight numbers for node indices in each line.

1) Read the element stiffness matrix elemK(24,24), 8 nodes * 3 degrees of freedom per node.
2) Read the number of elements, nel, and number of node, nnod, from the first line of the data file.
3) Number of degrees of freedom, ndof = nnod *3 - 300.
4) Allocate space for the sparse global matrix, matK(ndof, ndof).
5) Assembly:
Do (k = 1, nel)
Read eight node numbers from the k-th line, nodeindex(8);
Construct the index for degrees of freedom, dofindex(24):
Do (i = 1, 8)
Do (j = 1, 3)
dofindex((i - 1)*3 + j)= (nodeindex(i) - 1)*3 + j;
Use dofindex to assemble the element matrix elemK:
Do (i = 1, 24)
Do (j = 1, 24)
If (dofindex(i) < ndof AND dofindex(j) < ndof)
matK(dofindex(i), dofindex(j)) += elemK(i, j)

The input matrix contains a single column with <math>B(1) = 1</math>.
The output matrix takes the first three components of the state vector, that is, three displacements <math>UX</math>, <math>UY</math> and <math>UZ</math> for the first node.

The archive [http://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-assemble.tar.gz BoneModel-assemble.tar.gz] contains the element mass and stiffness matrices as well as the sample code in [[wikipedia:C++|C++]] to assemble the dynamic system.
The code can write the dynamic system in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format or can be used as a hook to transform the global matrices to an appropriate format.
The gzipped compressed data files for element assembly as described above can be downloaded from Table 1.

Model reduction for models <tt>BS010</tt> and <tt>BS10</tt> was performed in <ref name="rudnyi2005"/>. The benchmarking of the parallel [http://graal.ens-lyon.fr/MUMPS/ MUMPS] direct solver <ref name="amestoy2006"/> for the stiffness matrices is described in <ref name="rudnyi2006"/>.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
M \ddot{x}(t) + K x(t) &=& b \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>M \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}^{3 \times N}</math>.

System variants:

<tt>BS01</tt>: <math>N = 127224</math>,
<tt>BS10</tt>: <math>N = 914898</math>,
<tt>B010</tt>: <math>N = 986703</math>,
<tt>B025</tt>: <math>N = 2159661</math>,
<tt>B050</tt>: <math>N = 4934244</math>,
<tt>B120</tt>: <math>N = 11969688</math>.

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

@ARTICLE{RieWHetal95,
author = <nowiki>{B. van Rietbergen and H. Weinans and R. Huiskes and A. Odgaard}</nowiki>,
title = {A new method to determine trabecular bone elastic properties and loading using micromechanical finite-elements models},
journal = {Journal of Biomechanics},
volume = {28},
number = {1},
pages = {69--81},
year = {1995},
doi = {10.1016/0021-9290(95)80008-5}
}

==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="rietbergen1995">B. van Rietbergen, H. Weinans, R. Huiskes, A. Odgaard, [https://doi.org/10.1016/0021-9290(95)80008-5 A new method to determine trabecular bone elastic properties and loading using micromechanical finite-elements models] Journal of Biomechanics, 28(1): 69--81, 1995.</ref>

<ref name="rietbergen2003">B. van Rietbergen, R. Huiskes, F. Eckstein, P. Rueegsegger, [https://doi.org/10.1359/jbmr.2003.18.10.1781 Trabecular Bone Tissue Strains in the Healthy and Osteoporotic Human Femur], Journal of Bone and Mineral Research, 18(10): 1781--1787, 2003.</ref>

<ref name="lanyon1984">L.E. Lanyon, C.T. Rubin, [https://doi.org/10.1016/0021-9290(84)90003-4 Static versus dynamic loads as an influence on bone remodelling], Journal of Biomechanics, 17: 897--906, 1984.</ref>

<ref name="rudnyi2005">E.B. Rudnyi, B. van Rietbergen, J.G. Korvink. [http://repository.tue.nl/b3454bd9-f190-4c33-8dca-22c50b08edec Efficient Harmonic Simulation of a Trabecular Bone Finite Element Model by means of Model Reduction]. 12th Workshop "The Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields", Proceedings of the 12th FEM Workshop: 61--68, 2005</ref>

<ref name="amestoy2006">P.R. Amestoy, A. Guermouche and J.-Y. L'Excellent, S. Pralet, [https://doi.org/10.1016/j.parco.2005.07.004 Hybrid scheduling for the parallel solution of linear systems]. Parallel Computing, 32(2): 136--156, 2006.</ref>

<ref name="rudnyi2006">E.B. Rudnyi, B. van Rietbergen, J. G. Korvink. [http://modelreduction.com/doc/papers/rudnyi06tam.pdf Model Reduction for High Dimensional Micro-FE Models]. TAM'06, The Third HPC-Europa Transnational Access Meeting, Barcelona, 2006.</ref>

</references>

Silicon Nitride Membrane

2023-04-24T12:17:13Z

Lund: /* Dimensions */ typo

[[Category:benchmark]]
[[Category:Parametric]]
[[Category:linear]]
[[Category:first differential order]]
[[Category:time invariant]]

==Description==

<figure id="fig:tempprof">[[File:Fig_1.png|right|frame|<caption>silicon nitride membrane temperature profile</caption>]]</figure>

A '''silicon nitride membrane''' ([[wikipedia:Silicon_nitride|SiN]] membrane) <ref name="bechthold10"/> can be a part of a gas sensor,
but also a part of an infra-red sensor, a microthruster, an optical filter etc.
This structure resembles a microhotplate similar to other micro-fabricated devices such as gas sensors <ref name="spannhake05"/> and infrared sources <ref name="graf04"/> (See also [[Gas_Sensor|Gas Sensor Benchmark]]).
See Fig. 1, the temperature profile for the SiN membrane.

The governing heat transfer equation in the membrane is:

:<math> \nabla \cdot (\kappa \nabla T)+Q - \rho c_p \cdot \frac{\partial T}{\partial t}=0 </math>

where <math>\kappa </math> is the thermal conductivity in <math>W m^{-1} K^{-1}</math>,
<math>c_p</math> is the specific heat capacity in <math>J kg^{-1} K^{-1}</math>,
<math>\rho</math> is the mass density in <math>kg m^{-3}</math> and <math>T</math> is the temperature distribution.
We assume a homogeneous heat generation rate over a lumped resistor:

:<math>Q = \frac{u^2(t)}{R(T)}</math>

with <math>Q</math> the heat generation rate per unit volume in <math>W m^{-3}</math>.
We use the initial condition <math>T_0 = 273K</math>,
and the Dirichlet boundary condition <math>T = 273 K</math> at the bottom of the computational domain.

The convection boundary condition at the top of the membrane is

:<math>
q=h(T-T_{air}),
</math>

where <math>h</math> is the heat transfer coefficient between the membrane and the ambient air in <math>W m^{-2} K^{-1}</math>.

==Discretization==

Under the above convection boundary condition and assuming <math>T_{air}=0</math>,
a finite element discretization of the heat transfer model leads to the parametrized system as below,

:<math>
(E_0+\rho c_p \cdot E_1) \dot{T} +(A_0 +\kappa \cdot A_1 +h \cdot A_2)T = B \frac{u^2(t)}{R(T)}, \quad
y=C T,
</math>

where the volumetric heat capacity <math>\rho c_p</math>,
thermal conductivity <math>\kappa</math> and the heat transfer coefficient <math>h</math> between the membrane are kept as parameters.
The volumetric hear capacity <math>\rho c_p</math> is the product of two independent variables,
i.e. the specific hear capacity <math>c_p</math> and the density <math>\rho</math>.
The range of interest for the four independent variables are <math>\kappa \in [2, 5]</math>, <math>c_p \in [400, 750]</math>, <math>\rho \in [3000,3200]</math> and <math> h \in [10, 12]</math> respectively.
The frequency range is <math>f \in [0,25]Hz</math>.
What is of interest is the output in time domain.
The interesting time interval is <math>t \in [0,0.04]s</math>.

Here <math>R(T)</math> is either a constant heat resistivity <math>R(T)=R_0</math>, or <math>R(T)=R_0(1+\alpha T)</math>,
which depends linearly on the temperature.
Here we use <math>R_0=274.94 \Omega</math> and temperature coefficient <math>\alpha=2.293 \pm 0.006 \times 10^{-4}</math>.
The model was created and meshed in ANSYS. It contains a constant load vector <math>B</math> corresponding to the constant input power of <math>2.49mW</math>.
The number of degrees of freedom is <math>n=60,020</math>.

The input function <math>u(t)</math> is a step function with the value <math>1</math>,
which disappears at the time <math>0.02s</math>.
This means between <math>0s</math> and <math>0.02s</math> input is one and after that it is zero.
However, be aware that <math>u(t)</math> is just a factor with which the load vector B is multiplied and which corresponds to the heating power of <math>2.49mW</math>.
This means if one keeps <math>u(t)</math> as suggested above, the device is heated with <math>2.49mW</math> for the time length of 0.02s and after that the heating is turned off.
If for whatever reason, one wants the heating power to be <math>5mW</math>, then <math>u(t)</math> has to be set equal to two, etc.
When <math>R(T)=R_0(1+\alpha T)</math>, it is a function of the state vector <math>T</math> and hence, the system has non-linear input (It is also called a weakly nonlinear system.).

==Data==

The model is generated in ANSYS. The system matrices are in [http://math.nist.gov/MatrixMarket/ MatrixMarket] format and can be downloaded here: [[Media: SiN_membrane.tgz|SiN_membrane.tgz]].

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
(E_0 + \rho c_p E_1)\dot{x}(t) &=& -(A_0 + \kappa A_1 + h A_2)x(t) + Bu(t) \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>E_0 \in \mathbb{R}^{60020 \times 60020}</math>,
<math>E_1 \in \mathbb{R}^{60020 \times 60020}</math>,
<math>A_0 \in \mathbb{R}^{60020 \times 60020}</math>,
<math>A_1 \in \mathbb{R}^{60020 \times 60020}</math>,
<math>A_2 \in \mathbb{R}^{60020 \times 60020}</math>,
<math>B \in \mathbb{R}^{60020 \times 1}</math>,
<math>C \in \mathbb{R}^{2 \times 60020}</math>.

==References==

<references>

<ref name="bechthold10">T. Bechtold, D. Hohfeld, E. B. Rudnyi and M. Guenther, "[https://doi.org/10.1088/0960-1317/20/4/045030 Efficient extraction of thin-film thermal parameters from numerical models via parametric model order reduction]", J. Micromech. Microeng. 20(4): 045030, 2010.</ref>

<ref name="spannhake05">J. Spannhake, O. Schulz, A. Helwig, G. Müller and T. Doll, "[https://doi.org/10.1109/ICSENS.2005.1597811 Design, development and operational concept of an advanced MEMS IR source for miniaturized gas sensor systems]", Proc. Sensors: 762--765, 2005.</ref>

<ref name="graf04">M. Graf, D. Barrettino, S. Taschini, C. Hagleitner, A. Hierlemann and H. Baltes, "[https://doi.org/10.1021/ac035432h Metal oxide-based monolithic complementary metal oxide semiconductor gas sensor microsystem]", Anal. Chem., 76(15): 4437--4445, 2004.</ref>

</references>

==Contact==
'' [[User:Feng|Lihong Feng]] ''

[http://www.igs.uni-rostock.de/mitarbeiter/tamara-bechtold/ Tamara Bechtold]

Scanning Electrochemical Microscopy

2023-04-24T12:16:08Z

Lund: /* Dimensions */ D --> A

[[Category:benchmark]]
[[Category:parametric]]
[[Category:linear]]
[[Category:time varying]]
[[Category:Parametric]]
[[Category:first differential order]]
[[Category:nonzero initial condition]]

==Description==

<figure id="fig:cylin">
[[Image:Fig.1.JPG|thumb|right|300px|<caption>Cylindrical Electrode</caption>]]
</figure>

'''Scanning Electrochemical Microscopy''' (SECM) has many applications in current problems in the biological field.
Quantitative mathematical models have been developed for different operating modes of the SECM.
Except for some very specific problems, like the diffusion-controlled current on a circular electrode far away from the border,
solutions can only be obtained by numerical simulation, which is based on discretization of the model in space by an appropriate method like finite differences, finite elements, or boundary elements.
After discretization, a high-dimensional system of ordinary differential equations is obtained.
Its high dimensionality leads to high computational cost.

We consider a cylindrical electrode in Fig. 1.
The computation domain under the 2D-axisymmetric approximation includes the electrolyte under the electrode.
We assume that the concentration does not depend on the rotation angle.
A single chemical reaction takes place on the electrode:

:<math>Ox+e^-\Leftrightarrow Red, \quad \quad \quad \quad (1) </math>

where <math>Ox</math> and <math>Red</math> are two different species in the reaction.
According to the theory of SECM <ref name="mirkin01"/>, the species transport in the electrolyte is described by diffusion only.
The diffusion partial differential equation is given by the second [[wikipedia:Fick's_laws_of_diffusion|Fick's law]] as follows

:<math>
\frac{dc_1}{dt}=D_1\cdot \Delta ^2c_1 , \quad
\frac{dc_2}{dt}=D_2\cdot \Delta ^2c_2,
</math>

where <math>c_1</math> and <math>c_2</math> are the concentration fields of species <math>Ox</math> and <math>Red</math>, respectively.
The initial conditions are <math>c_1(0)=c_{1,0}</math> and <math>c_2(0)=c_{2,0}.</math> conditions at the glass and the bottom of the bath
are described by the Neumann boundary conditions of zero flux <math>\nabla c_1\cdot \vec{n}=0</math> and <math>\nabla c_2\cdot \vec{n}=0</math>.
Conditions at the border of the bulk are described by Dirichlet boundary conditions of constant concentration, equal to the initial conditions <math>c_1=c_{1,0}</math> and <math>c_2=c_{2,0}</math>.
The boundary conditions at the electrode are described by

:<math>
\nabla c_1\cdot \vec{n}=j, \,
\nabla c_2\cdot \vec{n}=-j. \quad \quad \quad \quad (2)
</math>

Here <math>j</math> is related to the forward reaction rate <math>k_f</math> and the backward reaction rate <math>k_b</math> through the [[wikipedia:Butler–Volmer_equation|Butler-Volmer equation]],

:<math>
j=k_f \cdot c_1-k_b \cdot c_2.
</math>

The reaction rates <math>k_f</math> and <math>k_b</math> are in the following form,

:<math>
k_f=k^0\exp{(\frac{\alpha z F(v(t)-v^0)}{RT})}, </math>

:<math>k_b=k^0\exp{(\frac{-(1-\alpha) z F(v(t)-v^0)}{RT})} .
</math>

Here, <math>k^0</math> is the heterogeneous standard rate constant, which is an empirical transmission factor for a heterogeneous reaction.
<math>F</math> is the [[wikipedia:Faraday_constant|Faraday-constant]], <math>R</math> is the [[wikipedia:Gas_constant|gas constant]], <math>T</math> is the temperature, and <math>z</math> is the number of exchanged electrons per reaction.
<math>u(t)=v(t)-v^0</math> is the difference between the electrode potential and the reference potential.
This difference, to which we refer below as voltage, changes during the measurement of a [[wikipedia:Voltammetry|voltammogram]].

==Model==

The control volume method has been used for the spatial discretization of (1).
Together with the boundary conditions, the resulting system of ordinary differential equations is as follows,

:<math>
E\frac{d\vec{c}}{dt}+K(u(t))\vec{c}-A\vec{c}=B,\quad
y(t)=C\vec{c},\quad
\vec{c}(0)=\vec{c}_0 \neq 0,
</math>

where E and <math>K(u(t))</math> are system matrices, <math>K(u(t))</math> is a function of voltage that in turn depends on time.
The voltage appears in the system matrix due to the boundary conditions (2).
The vector <math>\vec{c} \in \mathbb{R}^n</math> is the vector of unknown concentrations, which includes both the <math>Ox</math> and <math>Red</math> species.
The vector <math>B</math> is the load vector, which arises as a consequence of the Dirichlet boundary conditions imposed at the bulk boundary of the electrolyte. The total current is computed as an integral (sum) over the electrode surface.
The matrix <math>K(u(t))</math> has the following form,

:<math>
K(u(t))=K_1(u(t))+K_2(u(t)),
</math>

where <math>K_i(u(t))=h_i D_i, \, i=1,2,</math> and <math>h_1=\exp(\beta u(t)), \, h_2=\exp(-\beta u(t))</math>.
The voltage <math>u(t)</math> is a function of <math>\sigma</math>,

:<math>
u(t)=\sigma t-1, \, \text{for } t \leq \frac{2}{ \sigma}, \quad
u(t)=-\sigma t+3, \, \text{for } \frac{2}{ \sigma} < t \leq \frac{4}{ \sigma},
</math>

where <math>\sigma</math> can take four different values, <math>\sigma=0.5, \, 0.05, \, 0.005, \, 0.0005</math>. The constant <math>\beta</math> is computed from the parameters <math>\alpha, \, z, \, F, \, R,</math> and <math>T,</math> leading to the value <math>\beta=21.243036728240824</math>.
Although the system is a time-varying system, it can be considered as a parametrized system with two parameters <math>h_1</math> and <math>h_2</math>.

==Data==

The data of the system matrices <math>E, \ D_1, \ D_2, \ A, \ B, C</math> as well as the initial state <math>\vec{c}_0=x_0</math> are in MatrixMarket format (http://math.nist.gov/MatrixMarket/), and can be downloaded here [[Media:SECM.TGZ|SECM.tgz]]. The quantity of interest is the current which is computed by <math>I(t)=C(5,:)\vec{c}</math> in MATLAB notation. The associated plot is called the [[wikipedia:Cyclic_voltammetry|cyclic voltammogram]] <ref name="feng06"/>, which is the plot of the current changing with the voltage <math>u(t)</math>.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
E\dot{c}(t) &=& (A - h_1 A_1 - h_2 A_2)c(t) + Bu(t) \\
y(t) &=& Cc(t)
\end{array}
</math>

System dimensions:

<math>E \in \mathbb{R}^{16\,912 \times 16\,912}</math>,
<math>A \in \mathbb{R}^{16\,912 \times 16\,912}</math>,
<math>A_1 \in \mathbb{R}^{16\,912 \times 16\,912}</math>,
<math>A_2 \in \mathbb{R}^{16\,912 \times 16\,912}</math>,
<math>B \in \mathbb{R}^{16\,912 \times 1}</math>,
<math>C \in \mathbb{R}^{5 \times 16\,912}</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Scanning Electrochemical Microscopy'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Scanning_Electrochemical_Microscopy

@MISC{morwiki_secm,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Scanning Electrochemical Microscopy},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Scanning_Electrochemical_Microscopy}</nowiki>,
year = {2018}
}

* For the background on the benchmark:

@ARTICLE{morFenKRetal06,
author = {L. Feng, D. Koziol, E.B. Rudnyi, and J.G. Korvink},
title = {Parametric Model Reduction for Fast Simulation of Cyclic Voltammograms},
journal = {Sensor Letters},
volume = 4,
number = 2,
pages = {165--173},
year = 2006,
doi = {10.1166/sl.2006.021}
}

==References==

<references>

<ref name="mirkin01"> M.V. Mirkin, "[https://doi.org/10.1201/b11850-7 Chapter 5: Theory]", In: A.J. Bard and M.V. Mirkin, (eds.), Scanning Electrochemical Microscopy, CRC Press: 144--199, 2001.</ref>

<ref name="feng06"> L. Feng, D. Koziol, E.B. Rudnyi, and J.G. Korvink, "[https://doi.org/10.1166/sl.2006.021 Parametric Model Reduction for Fast Simulation of Cyclic Voltammograms]", Sensor Letters, 4(2): 165--173, 2006.</ref>

</references>

==Contact==

'' [[User:Feng|Lihong Feng]] ''

Modified Gyroscope

2023-04-20T19:50:59Z

Lund: update matrix names to match Models

[[Category:benchmark]]
[[Category:ODE]]
[[Category:linear]]
[[Category:Parametric]]
[[Category:affine parameter representation]]
[[Category:time invariant]]
[[Category:second differential order]]

==Description==
<figure id="fig:gyro">[[File:Gyroscope.jpg|300px|thumb|right|<caption>Schematic representation of the gyroscope.</caption>]]</figure>

The device is a [[wikipedia:Vibrating_structure_gyroscope#MEMS_gyroscopes|MEMS gyroscope]] based on the butterfly gyroscope <ref name="lienemann2004"/> developed at the [http://www.imego.com/ Imego institute] in Gothenburg,
Sweden (see also: [[Butterfly_Gyroscope|Butterfly Gyroscope]], where a non-parametrized model for the device is given).
A [[wikipedia:Gyroscope|gyroscope]] is a device used to measure angular rates in up to three axes.

The basic working principle of the '''gyroscope''' can be described as follows, see also <ref name="Moo07"/>.
Without applied external rotation, the paddles vibrate in phase with the function <math>z(t),</math> see Fig. 1.
Under the influence of an external rotation about the <math>x</math>-axis (drawn in red),
an additional force due to the Coriolis acceleration acts upon the paddles.
This force leads to an additional small out-of-phase vibration between two paddles on the same side of the bearing.
This out-of phase vibration is measured as the difference of the <math>z</math>-displacement of the nodes with the red dots.
Thus, measuring the displacement of two adjacent paddles, the rotation velocity can be ascertained.

==Motivation==

When planning for and making decisions on future improvements of the butterfly gyroscope, it is of importance to improve the efficiency of the gyro simulations. Repeated analysis 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 run time for repeated simulations.

==The Parametrized Model==

Two parameters are of special interest for the model.
The first one is the quantity that is to be sensed, the rotation velocity <math>\theta</math> around the <math>x</math>-axes.
The second parameter is the width of the bearing, <math>d</math>.
The parametrized system below is obtained by finite element discretization of the parametrized model (in the form of partial differential equations) for the '''gyroscope'''.
The details of constructing the parametrized system can be found in <ref name="Moo07"/>.
The system is of the following form:

:<math>
\begin{align}
M(d)\ddot{x}(t) +E(d,\theta)\dot{x}(t) +K(d)x(t) &= B, \\
y(t) &=Cx(t).
\end{align}
</math>

Here,

* <math>M(d)=M + dM_1\in \mathbb R^{n\times n}</math> is the mass matrix,
* <math>E(d,\theta)=\theta(E_1 + d E_2)\in \mathbb R^{n\times n}</math> is the damping matrix,
* <math>K(d)=K+(1/d)K_1+dK_2\in \mathbb R^{n\times n}</math> is the stiffness matrix,
* <math>B \in \mathbb R^{n \times 1}</math> is the load vector,
* <math>C \in \mathbb R^{1 \times n}</math> is the output matrix,
* <math>x \in \mathbb R^{n}</math> is the state vector,
* and <math>y \in \mathbb R </math> is the output response.

The quantity of interesting <math>y</math> of the system is <math>\delta z(t)</math>,
which is the difference of the displacement <math>z(t)</math> between the two red markings on the ''east'' side of the bearing (see <xr id="fig:gyro"/>).

The parameters of the system, <math>d</math> and <math>\theta</math>,
represent the width of the bearing(<math>d</math>) and the rotation velocity along the <math>x</math>-axis (<math>\theta</math>),
with the ranges: <math>\theta\in [10^{-7}, 10^{-5}]</math> and <math>d\in [1,2]</math>.

The device works in the frequency range <math>f \in [0.025, 0.25]</math>MHz and the degrees of freedom are <math>n = 17913</math>.

==Data==

The model is generated in ANSYS.
The system matrices <math>M, \, M_1, \, E_1, \, E_2, \, K, \, K_1, \,K_2</math>, and <math> B</math> are in the [http://math.nist.gov/MatrixMarket/ MatrixMarket format], and can be downloaded here: [[Media: Gyroscope_modi.tgz|Gyroscope_modi.tgz]].
The matrix <math>C</math> defines the output, which has zeros on all the entries, except on the 2315th entry, where the value is <math>-1</math>, and on the 5806th entry, the value is <math>1</math>;
in MATLAB notation, it is <tt>C(1,2315) = -1</tt> and <tt>C(1,5806) = 1</tt>.

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
(M + d M_1)\ddot{x}(t) + \theta(E_1 + d E_2) \dot{x}(t) + (K + d^{-1} K_1 + d K_2)x(t) &=& B \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

<math>M, M_1, E_1, E_2, K, K_1, K_2 \in \mathbb{R}^{17931 \times 17931}</math>,
<math>B \in \mathbb{R}^{17931 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 17931}</math>.

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_modgyro,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Modified Gyroscope},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Modified_Gyroscope}</nowiki>,
year = 2018
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morMoo07 morMoo07] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morMoo07 BibTeX])

==References==

<references>

<ref name="lienemann2004">J. Lienemann, D. Billger, E. B. Rudnyi, A. Greiner, J. G. Korvink, "[http://www.nsti.org/procs/Nanotech2004v2/6/W58.01 MEMS Compact Modeling Meets Model Order Reduction: Examples of the Application of Arnoldi Methods to Microsystem Devices]", TechConnect Briefs (Technical Proceedings of the 2004 NSTI Nanotechnology Conference and Trade Show, Volume 2): 303--306, 2004.</ref>

<ref name="Moo07">C. Moosmann, "[http://www.freidok.uni-freiburg.de/volltexte/3971/ ParaMOR Model Order Reduction for parameterized MEMS applications]", PhD thesis, Department of Microsystems Engineering, University of Freiburg, Freiburg, Germany 2007.</ref>

</references>

==Contact==

'' [[User:Feng]] ''

Models

2023-04-20T19:48:33Z

Lund: /* Affine-Parametric LTI-SOS (AP-LTI-SOS) */ fix matrix typos

[[Category:Benchmarks]]

==Benchmark Model Templates==
This page specifies templates for the types of models used as benchmark systems. In particular, the naming schemes established here are used in the corresponding data sets for all benchmarks. For example, <math>A</math> always serves as the name of the component matrix applied to the state <math>x(t)</math> in a linear time-invariant, first-order system.
For all models we assume an input <math>u : \mathbb{R} \to \mathbb{R}^m</math>, with components <math>u_j, j = 1, \ldots, m</math>,
a state <math>x : \mathbb{R} \to \mathbb{R}^n</math>,
and an output <math>y : \mathbb{R} \to \mathbb{R}^q</math>.
For all parametric models, we assume each component has <math>\ell</math> parameters; in cases where a component has fewer than <math>\ell</math> parameters, the extras are treated as <math>0</math>.
Some benchmarks (e.g., [[Bone Model]]) have a constant forcing term, in which case, it is assumed that <math>u(t)</math> is identically <math>1</math>.

===Linear Time-Invariant First-Order System (LTI-FOS)===

:<math>
\begin{align}
E\dot{x}(t) &= Ax(t) + Bu(t),\\
y(t) &= Cx(t) + Du(t),
\end{align}
</math>

with

<math>E \in \mathbb{R}^{n \times n}</math>,
<math>A \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>.

By default <math>E = I</math> and <math>D = 0</math>, unless explicitly provided.

===Linear Time-Varying First-Order System (LTV-FOS)===

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

with

<math>E : \mathbb{R} \to \mathbb{R}^{n \times n}</math>,
<math>A : \mathbb{R} \to \mathbb{R}^{n \times n}</math>,
<math>B : \mathbb{R} \to \mathbb{R}^{n \times m}</math>,
<math>C : \mathbb{R} \to \mathbb{R}^{q \times n}</math>,
<math>D : \mathbb{R} \to \mathbb{R}^{q \times m}</math>.

By default <math>E = I</math> and <math>D = 0</math>, unless explicitly provided.

===Affine-Parametric LTI-FOS (AP-LTI-FOS)===

:<math>
\begin{align}
(E + \sum_{i=1}^{\ell} p^E_i E_i)\dot{x}(t) &= (A + \sum_{i=1}^{\ell} p^A_i A_i) x(t) + (B + \sum_{i=1}^{\ell} p^B_i B_i)u(t),\\
y(t) &= (C + \sum_{i=1}^{\ell} p^C_i C_i)x(t),
\end{align}
</math>

with

<math>E, E_i \in \mathbb{R}^{n \times n}</math>;
<math>A, A_i \in \mathbb{R}^{n \times n}</math>;
<math>B, B_i \in \mathbb{R}^{n \times m}</math>; and
<math>C, C_i \in \mathbb{R}^{q \times n}</math>,
for all <math>i = 1, \ldots, \ell</math>.

By default <math>E = I, E_i = 0</math>, unless explicitly provided. If <math>A_i</math> are provided without <math>A</math>, then it is assumed <math>A = 0</math>. Likewise for <math>B</math>, <math>C</math>, and <math>E</math>.

===Linear Time-Invariant Second-Order System (LTI-SOS)===

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

with

<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 m}</math>,
<math>C_p, C_v \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>.

When <math>C_v = 0</math>, we denote <math>C = C_p</math>. By default <math>E = I</math> and <math>D = 0</math>, unless explicitly provided.

===Affine-Parametric LTI-SOS (AP-LTI-SOS)===

:<math>
\begin{align}
(M + \sum_{i=1}^{\ell} p^M_i M_i)\ddot{x}(t) + (E + \sum_{i=1}^{\ell} p^E_i E_i)\dot{x}(t) + (K + \sum_{i=1}^{\ell} p^K_i K_i)x(t) &= (B + \sum_{i=1}^{\ell} p^B_i B_i) u(t), \\
y(t) &= (C + \sum_{i=1}^{\ell} p^C_i C_i) x(t),
\end{align}
</math>

with

<math>M, M_i \in \mathbb{R}^{n \times n}</math>;
<math>E, E_i \in \mathbb{R}^{n \times n}</math>;
<math>K, K_i \in \mathbb{R}^{n \times n}</math>;
<math>B, B_i \in \mathbb{R}^{n \times m}</math>; and
<math>C, C_i \in \mathbb{R}^{q \times n}</math>,
for all <math>i = 1, \ldots, \ell</math>.

By default <math>E = I, E_i = 0</math>, unless explicitly provided. If <math>M_i</math> are provided without <math>M</math>, then it is assumed <math>M = 0</math>. Likewise for <math>K</math>, <math>B</math>, and <math>C</math>.

===Quadratic-Bilinear System (QBS)===

:<math>
\begin{align}
E\dot{x}(t) &= A x(t) + H x(t) \otimes x(t) + \sum_{j=1}^m N_j x(t) u_j(t) + B u(t), \\
y(t) &= Cx(t) + Du(t),
\end{align}
</math>

with

<math>E \in \mathbb{R}^{n \times n}</math>,
<math>A \in \mathbb{R}^{n \times n}</math>,
<math>H \in \mathbb{R}^{n \times n^2}</math>,
<math>N_j \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>.

===Nonlinear Time-Invariant First-Order System (NLTI-FOS)===

:<math>
\begin{align}
E\dot{x}(t) &= Ax(t) + Bu(t) + F f(x(t),u(t)),\\
y(t) &= Cx(t) + Du(t),
\end{align}
</math>

with

<math>E \in \mathbb{R}^{n \times n}</math>,
<math>A \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>,
<math>F \in \mathbb{R}^{n \times n}</math>,
<math>f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n</math>.

By default <math>F = I</math>, <math>E = I</math>, <math>D = 0</math>, unless explicitly provided.

===Nonlinear Time-Invariant Second-Order System (NLTI-SOS)===

:<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_p x(t) + C_v \dot{x}(t) + D u(t),
\end{align}
</math>

with

<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 m}</math>,
<math>F \in \mathbb{R}^{n \times n}</math>,
<math>C_p, C_v \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>,
<math>f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n</math>.

When <math>C_v = 0</math>, we denote <math>C = C_p</math>.

By default <math>F = I</math>, <math>E = I</math>, <math>D = 0</math>, unless explicitly provided.

===Other System Classes===
Affine-parametric and time-varying versions of nonlinear systems are clearly also possible by combining patterns of the above models.

Models

2023-04-20T19:45:07Z

Lund: update default conditions

[[Category:Benchmarks]]

==Benchmark Model Templates==
This page specifies templates for the types of models used as benchmark systems. In particular, the naming schemes established here are used in the corresponding data sets for all benchmarks. For example, <math>A</math> always serves as the name of the component matrix applied to the state <math>x(t)</math> in a linear time-invariant, first-order system.
For all models we assume an input <math>u : \mathbb{R} \to \mathbb{R}^m</math>, with components <math>u_j, j = 1, \ldots, m</math>,
a state <math>x : \mathbb{R} \to \mathbb{R}^n</math>,
and an output <math>y : \mathbb{R} \to \mathbb{R}^q</math>.
For all parametric models, we assume each component has <math>\ell</math> parameters; in cases where a component has fewer than <math>\ell</math> parameters, the extras are treated as <math>0</math>.
Some benchmarks (e.g., [[Bone Model]]) have a constant forcing term, in which case, it is assumed that <math>u(t)</math> is identically <math>1</math>.

===Linear Time-Invariant First-Order System (LTI-FOS)===

:<math>
\begin{align}
E\dot{x}(t) &= Ax(t) + Bu(t),\\
y(t) &= Cx(t) + Du(t),
\end{align}
</math>

with

<math>E \in \mathbb{R}^{n \times n}</math>,
<math>A \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>.

By default <math>E = I</math> and <math>D = 0</math>, unless explicitly provided.

===Linear Time-Varying First-Order System (LTV-FOS)===

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

with

<math>E : \mathbb{R} \to \mathbb{R}^{n \times n}</math>,
<math>A : \mathbb{R} \to \mathbb{R}^{n \times n}</math>,
<math>B : \mathbb{R} \to \mathbb{R}^{n \times m}</math>,
<math>C : \mathbb{R} \to \mathbb{R}^{q \times n}</math>,
<math>D : \mathbb{R} \to \mathbb{R}^{q \times m}</math>.

By default <math>E = I</math> and <math>D = 0</math>, unless explicitly provided.

===Affine-Parametric LTI-FOS (AP-LTI-FOS)===

:<math>
\begin{align}
(E + \sum_{i=1}^{\ell} p^E_i E_i)\dot{x}(t) &= (A + \sum_{i=1}^{\ell} p^A_i A_i) x(t) + (B + \sum_{i=1}^{\ell} p^B_i B_i)u(t),\\
y(t) &= (C + \sum_{i=1}^{\ell} p^C_i C_i)x(t),
\end{align}
</math>

with

<math>E, E_i \in \mathbb{R}^{n \times n}</math>;
<math>A, A_i \in \mathbb{R}^{n \times n}</math>;
<math>B, B_i \in \mathbb{R}^{n \times m}</math>; and
<math>C, C_i \in \mathbb{R}^{q \times n}</math>,
for all <math>i = 1, \ldots, \ell</math>.

By default <math>E = I, E_i = 0</math>, unless explicitly provided. If <math>A_i</math> are provided without <math>A</math>, then it is assumed <math>A = 0</math>. Likewise for <math>B</math>, <math>C</math>, and <math>E</math>.

===Linear Time-Invariant Second-Order System (LTI-SOS)===

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

with

<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 m}</math>,
<math>C_p, C_v \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>.

When <math>C_v = 0</math>, we denote <math>C = C_p</math>. By default <math>E = I</math> and <math>D = 0</math>, unless explicitly provided.

===Affine-Parametric LTI-SOS (AP-LTI-SOS)===

:<math>
\begin{align}
(M + \sum_{i=1}^{\ell} p^M_i M_i)\ddot{x}(t) + (E + \sum_{i=1}^{\ell} p^E_i E_i)\dot{x}(t) + (K + \sum_{i=1}^{\ell} p^K_i K_i)x(t) &= (B + \sum_{i=1}^{\ell} p^B_i B_i) u(t), \\
y(t) &= (C + \sum_{i=1}^{\ell} p^C_i C_i) x(t),
\end{align}
</math>

with

<math>M, M_i \in \mathbb{R}^{n \times n}</math>;
<math>E, E_i \in \mathbb{R}^{n \times n}</math>;
<math>K, K_i \in \mathbb{R}^{n \times n}</math>;
<math>B, B_i \in \mathbb{R}^{n \times m}</math>; and
<math>C, C_i \in \mathbb{R}^{q \times n}</math>,
for all <math>i = 1, \ldots, \ell</math>.

By default <math>E = I, E_i = 0</math>, unless explicitly provided. If <math>A_i</math> are provided without <math>A</math>, then it is assumed <math>A = 0</math>. Likewise for <math>B</math>, <math>C</math>, and <math>E</math>.

===Quadratic-Bilinear System (QBS)===

:<math>
\begin{align}
E\dot{x}(t) &= A x(t) + H x(t) \otimes x(t) + \sum_{j=1}^m N_j x(t) u_j(t) + B u(t), \\
y(t) &= Cx(t) + Du(t),
\end{align}
</math>

with

<math>E \in \mathbb{R}^{n \times n}</math>,
<math>A \in \mathbb{R}^{n \times n}</math>,
<math>H \in \mathbb{R}^{n \times n^2}</math>,
<math>N_j \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>.

===Nonlinear Time-Invariant First-Order System (NLTI-FOS)===

:<math>
\begin{align}
E\dot{x}(t) &= Ax(t) + Bu(t) + F f(x(t),u(t)),\\
y(t) &= Cx(t) + Du(t),
\end{align}
</math>

with

<math>E \in \mathbb{R}^{n \times n}</math>,
<math>A \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>,
<math>F \in \mathbb{R}^{n \times n}</math>,
<math>f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n</math>.

By default <math>F = I</math>, <math>E = I</math>, <math>D = 0</math>, unless explicitly provided.

===Nonlinear Time-Invariant Second-Order System (NLTI-SOS)===

:<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_p x(t) + C_v \dot{x}(t) + D u(t),
\end{align}
</math>

with

<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 m}</math>,
<math>F \in \mathbb{R}^{n \times n}</math>,
<math>C_p, C_v \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>,
<math>f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n</math>.

When <math>C_v = 0</math>, we denote <math>C = C_p</math>.

By default <math>F = I</math>, <math>E = I</math>, <math>D = 0</math>, unless explicitly provided.

===Other System Classes===
Affine-parametric and time-varying versions of nonlinear systems are clearly also possible by combining patterns of the above models.

Models

2023-04-20T19:42:35Z

Lund: /* Affine-Parametric LTI-SOS (AP-LTI-SOS) */ update defaults

[[Category:Benchmarks]]

==Benchmark Model Templates==
This page specifies templates for the types of models used as benchmark systems. In particular, the naming schemes established here are used in the corresponding data sets for all benchmarks. For example, <math>A</math> always serves as the name of the component matrix applied to the state <math>x(t)</math> in a linear time-invariant, first-order system.
For all models we assume an input <math>u : \mathbb{R} \to \mathbb{R}^m</math>, with components <math>u_j, j = 1, \ldots, m</math>,
a state <math>x : \mathbb{R} \to \mathbb{R}^n</math>,
and an output <math>y : \mathbb{R} \to \mathbb{R}^q</math>.
For all parametric models, we assume each component has <math>\ell</math> parameters; in cases where a component has fewer than <math>\ell</math> parameters, the extras are treated as <math>0</math>.
Some benchmarks (e.g., [[Bone Model]]) have a constant forcing term, in which case, it is assumed that <math>u(t)</math> is identically <math>1</math>.

===Linear Time-Invariant First-Order System (LTI-FOS)===

:<math>
\begin{align}
E\dot{x}(t) &= Ax(t) + Bu(t),\\
y(t) &= Cx(t) + Du(t),
\end{align}
</math>

with

<math>E \in \mathbb{R}^{n \times n}</math>,
<math>A \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>.

By default <math>E = I</math> and <math>D = 0</math>, unless explicitly provided.

===Linear Time-Varying First-Order System (LTV-FOS)===

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

with

<math>E : \mathbb{R} \to \mathbb{R}^{n \times n}</math>,
<math>A : \mathbb{R} \to \mathbb{R}^{n \times n}</math>,
<math>B : \mathbb{R} \to \mathbb{R}^{n \times m}</math>,
<math>C : \mathbb{R} \to \mathbb{R}^{q \times n}</math>,
<math>D : \mathbb{R} \to \mathbb{R}^{q \times m}</math>.

By default <math>E = I</math> and <math>D = 0</math>, unless explicitly provided.

===Affine-Parametric LTI-FOS (AP-LTI-FOS)===

:<math>
\begin{align}
(E + \sum_{i=1}^{\ell} p^E_i E_i)\dot{x}(t) &= (A + \sum_{i=1}^{\ell} p^A_i A_i) x(t) + (B + \sum_{i=1}^{\ell} p^B_i B_i)u(t),\\
y(t) &= (C + \sum_{i=1}^{\ell} p^C_i C_i)x(t),
\end{align}
</math>

with

<math>E, E_i \in \mathbb{R}^{n \times n}</math>;
<math>A, A_i \in \mathbb{R}^{n \times n}</math>;
<math>B, B_i \in \mathbb{R}^{n \times m}</math>; and
<math>C, C_i \in \mathbb{R}^{q \times n}</math>,
for all <math>i = 1, \ldots, \ell</math>.

By default <math>E = I, E_i = 0</math>, unless explicitly provided.

===Linear Time-Invariant Second-Order System (LTI-SOS)===

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

with

<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 m}</math>,
<math>C_p, C_v \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>.

When <math>C_v = 0</math>, we denote <math>C = C_p</math>. By default <math>E = I</math> and <math>D = 0</math>, unless explicitly provided.

===Affine-Parametric LTI-SOS (AP-LTI-SOS)===

:<math>
\begin{align}
(M + \sum_{i=1}^{\ell} p^M_i M_i)\ddot{x}(t) + (E + \sum_{i=1}^{\ell} p^E_i E_i)\dot{x}(t) + (K + \sum_{i=1}^{\ell} p^K_i K_i)x(t) &= (B + \sum_{i=1}^{\ell} p^B_i B_i) u(t), \\
y(t) &= (C + \sum_{i=1}^{\ell} p^C_i C_i) x(t),
\end{align}
</math>

with

<math>M, M_i \in \mathbb{R}^{n \times n}</math>;
<math>E, E_i \in \mathbb{R}^{n \times n}</math>;
<math>K, K_i \in \mathbb{R}^{n \times n}</math>;
<math>B, B_i \in \mathbb{R}^{n \times m}</math>; and
<math>C, C_i \in \mathbb{R}^{q \times n}</math>,
for all <math>i = 1, \ldots, \ell</math>.

By default <math>E = I, E_i = 0</math>, unless explicitly provided. If <math>E_i</math> are provided without <math>E</math>, then it is assumed <math>E = 0</math>.

===Quadratic-Bilinear System (QBS)===

:<math>
\begin{align}
E\dot{x}(t) &= A x(t) + H x(t) \otimes x(t) + \sum_{j=1}^m N_j x(t) u_j(t) + B u(t), \\
y(t) &= Cx(t) + Du(t),
\end{align}
</math>

with

<math>E \in \mathbb{R}^{n \times n}</math>,
<math>A \in \mathbb{R}^{n \times n}</math>,
<math>H \in \mathbb{R}^{n \times n^2}</math>,
<math>N_j \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>.

===Nonlinear Time-Invariant First-Order System (NLTI-FOS)===

:<math>
\begin{align}
E\dot{x}(t) &= Ax(t) + Bu(t) + F f(x(t),u(t)),\\
y(t) &= Cx(t) + Du(t),
\end{align}
</math>

with

<math>E \in \mathbb{R}^{n \times n}</math>,
<math>A \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>,
<math>F \in \mathbb{R}^{n \times n}</math>,
<math>f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n</math>.

By default <math>F = I</math>, <math>E = I</math>, <math>D = 0</math>, unless explicitly provided.

===Nonlinear Time-Invariant Second-Order System (NLTI-SOS)===

:<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_p x(t) + C_v \dot{x}(t) + D u(t),
\end{align}
</math>

with

<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 m}</math>,
<math>F \in \mathbb{R}^{n \times n}</math>,
<math>C_p, C_v \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>,
<math>f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n</math>.

When <math>C_v = 0</math>, we denote <math>C = C_p</math>.

By default <math>F = I</math>, <math>E = I</math>, <math>D = 0</math>, unless explicitly provided.

===Other System Classes===
Affine-parametric and time-varying versions of nonlinear systems are clearly also possible by combining patterns of the above models.

Models

2023-04-20T19:27:41Z

Lund: /* Benchmark Model Templates */ Add comments about defaults

[[Category:Benchmarks]]

==Benchmark Model Templates==
This page specifies templates for the types of models used as benchmark systems. In particular, the naming schemes established here are used in the corresponding data sets for all benchmarks. For example, <math>A</math> always serves as the name of the component matrix applied to the state <math>x(t)</math> in a linear time-invariant, first-order system.
For all models we assume an input <math>u : \mathbb{R} \to \mathbb{R}^m</math>, with components <math>u_j, j = 1, \ldots, m</math>,
a state <math>x : \mathbb{R} \to \mathbb{R}^n</math>,
and an output <math>y : \mathbb{R} \to \mathbb{R}^q</math>.
For all parametric models, we assume each component has <math>\ell</math> parameters; in cases where a component has fewer than <math>\ell</math> parameters, the extras are treated as <math>0</math>.
Some benchmarks (e.g., [[Bone Model]]) have a constant forcing term, in which case, it is assumed that <math>u(t)</math> is identically <math>1</math>.

===Linear Time-Invariant First-Order System (LTI-FOS)===

:<math>
\begin{align}
E\dot{x}(t) &= Ax(t) + Bu(t),\\
y(t) &= Cx(t) + Du(t),
\end{align}
</math>

with

<math>E \in \mathbb{R}^{n \times n}</math>,
<math>A \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>.

By default <math>E = I</math> and <math>D = 0</math>, unless explicitly provided.

===Linear Time-Varying First-Order System (LTV-FOS)===

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

with

<math>E : \mathbb{R} \to \mathbb{R}^{n \times n}</math>,
<math>A : \mathbb{R} \to \mathbb{R}^{n \times n}</math>,
<math>B : \mathbb{R} \to \mathbb{R}^{n \times m}</math>,
<math>C : \mathbb{R} \to \mathbb{R}^{q \times n}</math>,
<math>D : \mathbb{R} \to \mathbb{R}^{q \times m}</math>.

By default <math>E = I</math> and <math>D = 0</math>, unless explicitly provided.

===Affine-Parametric LTI-FOS (AP-LTI-FOS)===

:<math>
\begin{align}
(E + \sum_{i=1}^{\ell} p^E_i E_i)\dot{x}(t) &= (A + \sum_{i=1}^{\ell} p^A_i A_i) x(t) + (B + \sum_{i=1}^{\ell} p^B_i B_i)u(t),\\
y(t) &= (C + \sum_{i=1}^{\ell} p^C_i C_i)x(t),
\end{align}
</math>

with

<math>E, E_i \in \mathbb{R}^{n \times n}</math>;
<math>A, A_i \in \mathbb{R}^{n \times n}</math>;
<math>B, B_i \in \mathbb{R}^{n \times m}</math>; and
<math>C, C_i \in \mathbb{R}^{q \times n}</math>,
for all <math>i = 1, \ldots, \ell</math>.

By default <math>E = I, E_i = 0</math>, unless explicitly provided.

===Linear Time-Invariant Second-Order System (LTI-SOS)===

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

with

<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 m}</math>,
<math>C_p, C_v \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>.

When <math>C_v = 0</math>, we denote <math>C = C_p</math>. By default <math>E = I</math> and <math>D = 0</math>, unless explicitly provided.

===Affine-Parametric LTI-SOS (AP-LTI-SOS)===

:<math>
\begin{align}
(M + \sum_{i=1}^{\ell} p^M_i M_i)\ddot{x}(t) + (E + \sum_{i=1}^{\ell} p^E_i E_i)\dot{x}(t) + (K + \sum_{i=1}^{\ell} p^K_i K_i)x(t) &= (B + \sum_{i=1}^{\ell} p^B_i B_i) u(t), \\
y(t) &= (C + \sum_{i=1}^{\ell} p^C_i C_i) x(t),
\end{align}
</math>

with

<math>M, M_i \in \mathbb{R}^{n \times n}</math>;
<math>E, E_i \in \mathbb{R}^{n \times n}</math>;
<math>K, K_i \in \mathbb{R}^{n \times n}</math>;
<math>B, B_i \in \mathbb{R}^{n \times m}</math>; and
<math>C, C_i \in \mathbb{R}^{q \times n}</math>,
for all <math>i = 1, \ldots, \ell</math>.

By default <math>E = I, E_i = 0</math>, unless explicitly provided.

===Quadratic-Bilinear System (QBS)===

:<math>
\begin{align}
E\dot{x}(t) &= A x(t) + H x(t) \otimes x(t) + \sum_{j=1}^m N_j x(t) u_j(t) + B u(t), \\
y(t) &= Cx(t) + Du(t),
\end{align}
</math>

with

<math>E \in \mathbb{R}^{n \times n}</math>,
<math>A \in \mathbb{R}^{n \times n}</math>,
<math>H \in \mathbb{R}^{n \times n^2}</math>,
<math>N_j \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>.

===Nonlinear Time-Invariant First-Order System (NLTI-FOS)===

:<math>
\begin{align}
E\dot{x}(t) &= Ax(t) + Bu(t) + F f(x(t),u(t)),\\
y(t) &= Cx(t) + Du(t),
\end{align}
</math>

with

<math>E \in \mathbb{R}^{n \times n}</math>,
<math>A \in \mathbb{R}^{n \times n}</math>,
<math>B \in \mathbb{R}^{n \times m}</math>,
<math>C \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>,
<math>F \in \mathbb{R}^{n \times n}</math>,
<math>f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n</math>.

By default <math>F = I</math>, <math>E = I</math>, <math>D = 0</math>, unless explicitly provided.

===Nonlinear Time-Invariant Second-Order System (NLTI-SOS)===

:<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_p x(t) + C_v \dot{x}(t) + D u(t),
\end{align}
</math>

with

<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 m}</math>,
<math>F \in \mathbb{R}^{n \times n}</math>,
<math>C_p, C_v \in \mathbb{R}^{q \times n}</math>,
<math>D \in \mathbb{R}^{q \times m}</math>,
<math>f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n</math>.

When <math>C_v = 0</math>, we denote <math>C = C_p</math>.

By default <math>F = I</math>, <math>E = I</math>, <math>D = 0</math>, unless explicitly provided.

===Other System Classes===
Affine-parametric and time-varying versions of nonlinear systems are clearly also possible by combining patterns of the above models.