MOR Wiki - User contributions [en]

Transmission Lines

2022-03-24T13:55:35Z

Mitchell: ' versus ^T typo

[[Category:benchmark]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:DAE index one]]
==Definition==

In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency,
that is, currents with a frequency high enough that their wave nature must be taken into account.
'''Transmission lines''' are used for purposes such as connecting radio transmitters and receivers with their antennas,
distributing cable television signals, and computer network connections.

In many electric circuits, the length of the wires connecting the components can often be ignored.
That is, the voltage on the wire at a given time can be assumed to be the same at all points.
However, when the voltage changes as fast as the signal travels through the wire,
the length becomes important and the wire must be treated as a transmission line, with distributed parameters.
Stated in another way, the length of the wire is important when the signal includes frequency components with corresponding wavelengths comparable to or less than the length of the wire.

A common rule of thumb is that the cable or wire should be treated as a transmission line if its length is greater than <math>1/10</math> of the wavelength,
and the interconnect is called "electrically long".
At this length the phase delay and the interference of any reflections on the line (as well as other undesired effects) become important and can lead to unpredictable behavior in systems which have not been carefully designed using transmission line theory.



==Description==

The decrease of integrated circuit feature size and the increase of operating frequencies require 3-D electromagnetic methods,
such as the [[wikipedia:Partial_element_equivalent_circuit|Partial Element Equivalent Circuit]] (PEEC) method; it stems from the integral equation form of [[wikipedia:Maxwell's_equations|Maxwell's equations]].
The main difference of the PEEC method with other integral-Equation-based techniques, such as the method of moments, resides in the fact that it provides a circuit interpretation of the [[wikipedia:Electric-field_integral_equation|Electric Field Integral Equation]] (EFIE) in terms of partial elements, namely resistances, partial inductances, and coefficients of potential. In the standard approach, volumes and surfaces are discretized into elementary regions, hexahedra, and patches respectively over which the current and charge densities are expanded into a series of basis functions.
Pulse basis functions are usually adopted as expansion and weight functions.
Such choice of pulse basis functions corresponds to assuming constant current density and charge density over the elementary volume (inductive) and surface (capacitive) cells, respectively.
Following the standard Galerkin's testing procedure, topological elements, namely nodes and branches, are generated and electrical lumped elements are identified modeling both the magnetic and electric field coupling.
Conductors are modeled by their ohmic resistance, while dielectrics requires modeling the excess charge due to the dielectric polarization.
Magnetic and electric field coupling are modeled by partial inductances and coefficients of potential, respectively.

The magnetic field coupling between two inductive volume cells <math>\alpha</math> and <math>\beta</math> is described by the partial inductance

:<math> L_{p_{\alpha\beta}}=\frac{\mu}{4\pi}\frac{1}{a_{\alpha}a_{\beta}}\int_{u_{\alpha}}\int_{u_{\beta}}\frac{1}{R_{\alpha\beta}}\,du_{\alpha}\,du_{\beta} </math>

where <math>R_{\alpha\beta}</math> is the distance between any two points in the volumes <math>u_{\alpha}</math> and <math>u_{\beta}</math> with <math>a_{\alpha}</math> and <math>a_{\beta}</math> their cross section. The electric field coupling between two capacitive surface cells <math>\delta</math> and <math>\gamma</math> is modeled by the coefficient of the potential

:<math> P_{\delta\gamma}=\frac{1}{4\pi\epsilon}\frac{1}{S_{\delta}S_{\gamma}}\int_{S_{\delta}}\int_{S_{\gamma}}\frac{1}{R_{\delta\gamma}}\,dS_{\delta}\,dS_{\gamma} </math>

where <math>R_{\delta\gamma}</math> is the distance between any two points on the surfaces <math>\delta</math> and <math>\gamma</math>, while <math>S_{\delta}</math> and <math>S_{\gamma}</math> denote the area of their respective surfaces <ref name="ferranti11"/>.

Generalized Kirchhoff's laws for conductors, when dielectrics are considered, can be rewritten as

:<math> \textbf{P}^{-1}\frac{d\textbf{v}(t)}{dt}-\textbf{A}^T\textbf{i}(t)+\textbf{i}_e(t)=0, </math>

<figure id="fig:peec">[[File:Peec.jpg|400px|frame|<caption>Illustration of PEEC circuit electrical quantities for a conductor elementary cell (Figure from <ref name="ferranti11"/>).</caption>]]</figure>

:<math> -\textbf{A}\textbf{v}(t)-\textbf{L}_p\frac{d\textbf{i}(t)}{dt}-\textbf{v}_d(t)=0, </math>

:<math> \textbf{i}(t)=\textbf{C}_d\frac{d\textbf{v}_d(t)}{dt} </math>

where <math>\textbf{A}</math> is the connectivity matrix, <math>\textbf{v}(t)</math> denotes the node potentials to infinity, <math>\textbf{i}(t)</math> and <math>\textbf{i}_e(t)</math> represent the currents flowing in volume cells and the external currents, respectively, <math>\textbf{v}_d(t)</math> is the excess capacitance voltage drop, which is related to the excess charge by <math>\textbf{v}_d(t)=\textbf{C}_d^{-1}\textbf{q}_d(t)</math>. A selection matrix <math>\textbf{K}</math> is introduced to define the port voltages by selecting node potentials. The same matrix is used to obtain the external currents <math>\textbf{i}_e(t)</math> by the currents <math>\textbf{i}_s(t)</math>, which are of opposite sign with respect to the <math>n_p</math> port currents <math>\textbf{i}_p(t)</math>,

:<math> \textbf{v}_p(t)=\textbf{K}\textbf{v}(t), </math>

:<math> \textbf{i}_e(t)=\textbf{K}^T\textbf{i}_s(t). </math>

An example of PEEC circuit electrical quantities for a conductor elementary cell is illustrated, in the Laplace domain, in Fig. 1, where the current-controlled voltage sources <math>sL_{p,ij}I_j</math> and the current-controlled current sources <math>I_{cci}</math> model the magnetic and electric coupling, respectively.

Thus, assuming that we are interested in generating an admittance representation having <math>n_p</math> output currents under voltage excitation, and let us denote with <math>n_n</math> the number of nodes, <math>n_i</math> the number of branches where currents flow, <math>n_c</math> the number of branches of conductors, <math>n_d</math> the number of dielectrics, <math>n_d</math> the additional unknowns since dielectrics require the excess capacitance to model the polarization charge, and <math>n_u=n_i+n_d+n_n+n_p</math> the global number of unknowns, and if the [[wikipedia:Modified_nodal_analysis|Modified Nodal Analysis]] (MNA) approach is used, we have:

:::<math> \left[ \begin{array}{cccc} \textbf{P} & \textbf{0}_{n_n,n_i} & \textbf{0}_{n_n,n_d} & \textbf{0}_{n_n,n_p} \\ \textbf{0}_{n_i,n_n} & \textbf{L}_p & \textbf{0}_{n_i,n_d} & \textbf{0}_{n_i,n_p} \\ \textbf{0}_{n_d,n_n} & \textbf{0}_{n_d,n_i} & \textbf{C}_d & \textbf{0}_{n_d,n_p} \\ \textbf{0}_{n_p,n_n} & \textbf{0}_{n_p,n_i} & \textbf{0}_{n_p,n_d} & \textbf{0}_{n_p,n_p} \end{array}\right]\frac{d}{dt}\left[ \begin{array}{c}\textbf{q}(t) \\ \textbf{i}(t) \\ \textbf{v}_d(t) \\ \textbf{i}_s(t) \end{array}\right]= </math>

:<math>= - \left[ \begin{array}{cccc}\textbf{0}_{n_n,n_n} & -\textbf{P}\textbf{A}^T & \textbf{0}_{n_n,n_d} & \textbf{P}\textbf{K}^T \\ \textbf{AP} & \textbf{R} & \Phi & \textbf{0}_{n_i,n_p} \\ \textbf{0}_{n_d,n_n} & -\Phi^T & \textbf{0}_{n_d,n_d} & \textbf{0}_{n_d,n_p} \\ -\textbf{K}\textbf{P} & \textbf{0}_{n_p,n_i} & \textbf{0}_{n_p,n_d} & \textbf{0}_{n_p,n_p} \end{array}\right]\cdot\left[ \begin{array}{c} \textbf{q}(t) \\ \textbf{i}(t) \\ \textbf{v}_d(t) \\ \textbf{i}_s(t) \end{array}\right]+ \left[ \begin{array}{c}\textbf{0}_{n_n+n_i+n_d,n_p} \\ -\textbf{I}_{n_p,n_p} \end{array}\right] \cdot [ \textbf{v}_p(t) ]. </math>

Here <math>\textbf{0}</math> is a matrix of zeros, <math>\textbf{I}</math> is the identity matrix, both are with appropriate dimensions, and <math>\Phi=\left[ \begin{array}{c} \textbf{0}_{n_c,n_d} \\ \textbf{I}_{n_d,n_d} \end{array}\right]</math>. Then, in a more compact form, the above equation can be written as:

:<math> \left\{ \begin{array}{c} \textbf{C}\frac{d\textbf{x}(t)}{dt}=-\textbf{G}\textbf{x}(t)+\textbf{B}\textbf{u}(t)\\
\textbf{i}_p(t)=\textbf{L}^T\textbf{x}(t) \end{array}\right . \qquad (1)</math>

with <math>\textbf{x}(t)=\left[ \begin{array}{cccc} \textbf{q}(t)\quad\textbf{i}(t)\quad\textbf{v}_d(t)\quad\textbf{i}_s(t) \end{array}\right]^T</math>. Since this is an <math>n_p</math>-port formulation, whereby the only sources are the voltage sources at the <math>n_p</math>-ports nodes, <math>\textbf{B}=\textbf{L}</math> where <math>\textbf{B}\in\mathbb R^{n_u\times n_p}</math> (for more details on this model, refer to <ref name="ferranti11"/>).

==Motivation of MOR==

Since the number of equations produced by 3-D electromagnetic method PEEC is usually very large,
the inclusion of the PEEC model directly into a circuit simulator (like [[wikipedia:SPICE|SPICE]]) is computationally intractable for complex structures,
where the number of circuit elements can be tens of thousands.

==Data==

All data sets (in a MATLAB formatted data, downloadable in [[Media:TransmissionLines.rar|TransmissionLines.rar]]) in Fig. 1 are referred to as the multiconductor '''transmission lines''' in a MNA form, coming from the PEEC method (then, with dense matrices since they are obtained from the integral formulation of Maxwell's equation).
The LTI descriptor systems have the form of, equation <math>(1)</math>, where <math>C\in\mathbb R^{n\times n}</math> (with <math>C=C^T\ge0</math>), <math>G\in\mathbb R^{n\times n}</math> (with <math>G+G^T\ge0</math>), <math>B\in\mathbb R^{n\times m}</math> and <math>L=B^T</math>, <math>x(t)\in\mathbb R^n</math> is the vector of variables (charges, currents and node potential), the input signal <math>u(t)\in\mathbb R^m</math> are the sources (current or voltage generators depending on what one wants to analyze:
the impedances or the admittances) linked to some node, the output <math>y(t)\in\mathbb R^m</math> are the observation across the node where the sources are inserted. An accurate model of the dynamics of these data sets is generated between 10 kHz and 20 GHz.

<figtable id="tab:peec">
{| border="1"
! Name of the data set !! Matrices !! Dimension !! Number of inputs
|-
| <tt>dsysPEEC-MTLn1600m14</tt> || <math>G</math>, <math>B</math>, <math>C</math> (<math>L=B^T</math>) || 1600 || 14
|-
| <tt>dsysPEEC-MTLn2654m30</tt> || dss object (*) || 2654 || 30
|-
| <tt>dsysPEEC-MTLn5248m62</tt> || dss object (*) || 5248 || 62
|}
</figtable>

one can extract the matrices with Matlab command:

[G,B,L,D,C] = dssdata(dssObjectName);

e.g., if one wants to work on one of the last two data sets of this table,
just load it into the Matlab Workspace and type the command aforementioned on the Command Windows;
for the first example, once one loads the data, the Workspace shows directly the matrices.
Note that <math>D = 0</math>.

==Dimensions==

System structure:

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

System dimensions:

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

System variants:

<tt>dsysPEEC-MTLn1600m14</tt>: <math>N = 1600</math>, <math>M = 14</math>,
<tt>dsysPEEC-MTLn2654m30</tt>: <math>N = 2654</math>, <math>M = 30</math>,
<tt>dsysPEEC-MTLn5248m62</tt>: <math>N = 5248</math>, <math>M = 62</math>

==References==

<references>

<ref name="ferranti11"> F. Ferranti, G. Antonini, T. Dhaene, L. Knockaert, and A. E. Ruehli, "[https://doi.org/10.1109/TCPMT.2010.2101912 Physics-based passivity-preserving parameterized model order reduction for PEEC circuit analysis]", IEEE Transactions on Components, Packaging and Manufacturing Technology, 1(3): 399--409, 2011.</ref>

</references>

==Contact==

[[User:Deluca]]

[[User:Feng]]

Mass-Spring-Damper

2022-03-24T13:54:28Z

Mitchell: x(t) instead of x instead f

{{preliminary}} 

[[Category:benchmark]]
[[Category:nonlinear]]
[[Category:ODE]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:second differential order]]
[[Category:Procedural]]
[[Category:SISO]]

==Description: Mass-Spring-Damper System==

This benchmark is a generalization of the nonlinear [[wikipedia:Mass-spring-damper_model|mass-spring-damper]] system presented in <ref name="kawano15"/>,
which is concerned with modeling the a mechanical systems consisting of chained masses, linear and nonlinear springs, and dampers.
The underlying mathematical model is a second order system:
:<math>
\begin{align}
M \ddot{x}(t) + D \dot{x}(t) + K x(t) + f(x(t)) &= B u(t), \\
y(t) &= C x(t).
\end{align}
</math>

===First Order Representation===

The second order system can be represented as a first order system as follows:

:<math>
\begin{align}
\begin{pmatrix} 1 & 0 \\ 0 & M \end{pmatrix} \begin{pmatrix} \dot{p} \\ \dot{v} \end{pmatrix} &=
\begin{pmatrix} 0 & 1 \\ K & D \end{pmatrix} \begin{pmatrix} p \\ v \end{pmatrix} +
\begin{pmatrix} 0 \\ f_p(p) \end{pmatrix} + \begin{pmatrix} 0 \\ B_v \end{pmatrix} \\
y &= \begin{pmatrix} C_p & 0 \end{pmatrix} \begin{pmatrix} p \\ v \end{pmatrix}
\end{align}
</math>

with the components:

:<math>
M = m \begin{pmatrix} 1 \\ & \ddots \end{pmatrix}, \quad
K_0 = k_l \begin{pmatrix} -2 & 1 \\ 1 & -2 & \ddots \\ & \ddots & \ddots \end{pmatrix}, \quad
D = d \begin{pmatrix} -2 & 1 \\ 1 & -2 & \ddots \\ & \ddots & \ddots \end{pmatrix}, \quad
B_v = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 \end{pmatrix}, \quad
C_p = \begin{pmatrix} 0 & \dots & 0 & 1 \end{pmatrix},
</math>

and the nonlinear term:

:<math>
f_p(p) = -k_n \Big( \begin{pmatrix} 1 & -1 \\ & \ddots & \ddots \end{pmatrix} p \Big)^3 -k_n \Big( \begin{pmatrix} 1 \\ -1 & \ddots \\ & \ddots \end{pmatrix} p \Big)^3
</math>

and thus yielding the classic first order components:

:<math>
E = \begin{pmatrix} 1 & 0 \\ 0 & M \end{pmatrix}, \quad
A = \begin{pmatrix} 0 & 1 \\ K & D \end{pmatrix}, \quad
B = \begin{pmatrix} 0 \\ B_v \end{pmatrix}, \quad
C = \begin{pmatrix} C_p & 0 \end{pmatrix}.
</math>

The parameters for the mass <math>m</math>, linear spring constant <math>k_l</math>, nonlinear spring constant <math>k_n</math>, and damping <math>d</math> are chosen in <ref name="kawano15"/> as <math>m=1</math>, <math>k_l=1</math>, <math>k_n=2</math>, and <math>d=1</math>.

==Data==

The following Matlab code assembles the above described <math>E</math>, <math>A</math>, <math>B</math> and <math>C</math> parameter dependent matrices and the function <math>f</math> for a given number of subsystems <math>N</math>.

<div class="thumbinner" style="width:540px;text-align:left;">
<source lang="matlab">
function [E,A,B,C,f] = msd(N)

U = speye(N); % Sparse unit matrix
T = gallery('tridiag',N,-1,2,-1); % Sparse tridiagonal matrix
H = gallery('tridiag',N,-1,1,0); % Sparse transport matrix
Z = sparse(N,N); % Sparse all zero matrix
z = sparse(N,1); % Sparse all zero vector

E = @(m) [U,Z;Z,m*U]; % Handle to parametric E matrix
A = @(kl,d) [Z,U;kl*T,d*T]; % Handle to parametric A matrix
B = sparse(2*N,1,1,2*N,1);
C = sparse(N,1,1,2*N,1);
f = @(x,kn) [z;-kn*( (H'*x(N+1:end)).^3 - (H*x(N+1:end)).^3)];
end
</source>
</div>

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
E(m) \dot{x}(t) &=& A(k_l,d)x(t) + f(x(t);k_n) + Bu(t) \\
y(t) &=& Cx(t)
\end{array}
</math>

System dimensions:

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

==Citation==

To cite this benchmark, use the following references:

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

@MISC{morwiki_msd,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Mass-Spring-Damper System},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Mass-Spring-Damper}</nowiki>,
year = 2018
}

* For the background on the benchmark:

@INPROCEEDINGS{morKawS15,
title = {Model Reduction by Generalized Differential Balancing},
author = {Y. Kawano and J.M.A. Scherpen},
booktitle = {Mathematical Control Theory I: Nonlinear and Hybrid Control Systems},
series = {Lecture Notes in Control and Information Sciences},
volume = {461},
pages = {349--362},
year = {2015},
doi = {10.1007/978-3-319-20988-3}
}

==References==

<references>

<ref name="kawano15">Y. Kawano and J.M.A. Scherpen, [https://doi.org/10.1007/978-3-319-20988-3_19 Model Reduction by Generalized Differential Balancing], In: Mathematical Control Theory I: Nonlinear and Hybrid Control Systems, Lecture Notes in Control and Information Sciences 461: 349--362, 2015.</ref>

</references>

Submission rules

2022-03-24T10:41:50Z

Mitchell: tried to fix broken sentence

[[Category:Miscellaneous]]


The MOR Wiki [[:Category:Benchmark|benchmark collection]] contains benchmarks related to model order reduction. Its goal is to supply the dynamical systems in a computer-readable format to researchers from different areas. They can then test new algorithms, software, etc.

The collection contains documents and wikipages that are supposed to be processed by a human being and data files that are supposed to be read automatically by software. As a result, with minor exceptions, the rules concerning documents have a recommendation status, and the rules concerning data files are obligatory.

We start with describing the benchmark creation process, then we present the description of what content is needed on the wikipages as well as the supplement documents, and finally we describe the data files.

==Benchmark Creation==
Anyone who has an example that could be relevant for this MOR benchmark collection can add it to the collection, after the editors decide that it is suitable for our collection. In order to be allowed to create a benchmark, one has to write an email to the editors asking for access to the wiki describing the benchmark to be created. If the editors accept the benchmark they grant access to the authors and create an empty page for the benchmark. The submitter needs to then edit the created wikipage, upload the data files and supplementary documents and link them within the page. See 2.1 for the content requirement.

==Content Rules==

The supplementary documents as well as the wikipage may be written by different authors, however each individual document should be written according to conventional scientific practice, that is, it describes the matters in such a way that, at least in principle, anyone could reproduce the results presented. The authors should understand that the document may be read by people from quite different disciplines. Hence, abbreviations should be avoided or at least explained and references to the background ideas should be made.

===Content Requirement===
Most of this content can be included in the wiki page directly. However, this is not necessary.
One should describe the origin of the dynamic system and its relevance to the application area. It is important to present the mathematical model, the meaning of the inputs and outputs, and the desired behavior from the application viewpoint.

The following points should be considered and if possible included in your benchmark:

* The purpose of the model should be explained clearly. (For instance, simulation, iterative system design, feedback control design, ...)
* Why should the model be reduced at all? (For instance, reducing simulation time, reducing implementation effort in observers, controllers...)
* What are the QUALITATIVE requirements to the reduced model? What variables are to be approximated well? Is the step response to be approximated or is it the bode plot? What are typical input signals? (Some systems are driven by a step function and nothing else, others are driven by a wide variety of input signals, others are used in closed loops and can cause instabilities, although being stable themselves.)
* What are the QUANTITATIVE requirements to the reduced model? Best would be if the authors of any individual model can suggest some cost functions (performance indices) to be used for the comparison. These can be in time domain, or in frequency domain (including bandwidth), or both.
* Are there limits of input and state variables known? (application related or generally)? What are the physical limits where the model becomes useless/false? If known a-priori: Out of the technically typical input signals, which one will cause "the most nonlinear" behavior?
* How many parameters are included in the model, are they physical parameters or geometrical parameters.
* Is the system linear or nonlinear? Is it a first or a second order system?
* When the model is used to test, e.g. an algorithm, can some of the parameters in the model be fixed, such that the model includes fewer parameters, which makes the testing easier?
* If possible, the source code which generates the model is encouraged to be provided.

* We stress the importance to describe the software employed, as well as its related options. For example, if the model is generated from ANSYS, the software ANSYS is better to be mentioned. If necessary, please also describe some related options like boundary conditions, initial conditions, and other parameters which need to be assigned when generating the model. Please go to the benchmark webpage [[Silicon nitride membrane]] for a reference.

* If the dynamic system is obtained from partial differential equations, then the information about material properties, geometrical data, initial and boundary conditions should be given. The exception to this rule is the case when the original model comes from industry. In this case, if trade secrets are tied with the information mentioned, it may be kept hidden.

* The authors are encouraged to produce several dynamic models of different dimensions in order to provide an opportunity to apply different software and to research scalability issues. If an author has an interactive page on his/her server to generate benchmarks, a link to this page is welcome.

* The dynamic system may be obtained by means of compound matrices, for example, when the second-order system is converted to first-order. In this case, the document should describe such a transformation but in the data file the original and not the compound matrices should be given. This allows to research other ways of model reduction of the original system.

The above information is in a general sense. Therefore, some items may not be applicable to certain benchmarks. However, it is recommended that the items that apply to the benchmark provided should be fully considered.

===Additional Content===

This will typically be given in form of a document which we restrict to the following types: *.pdf, *.gif, *.jpeg, *.jpg, *.png,. This documents should include author names.

# The solution of the original benchmark that contains sample outputs for the usual input signals. Plots and numerical values of time and frequency response. Eigenvalues and eigenvectors, singular values, poles, zeros, etc.
# Model reduction and its results as compared to the original system.
# Description of any other related results.
# We stress the importance of describing the software employed as well as its related options.

==Data Files==

All the numerical data for the collection can be considered as a list of matrices, a vector being a m x 1 matrix. As a result, there should be a naming convention for the matrices.

===Naming convention===

Below <math>x</math> is a state vector, <math>\mu</math> is a vector of parameters in the system, if any.

For the two cases of a linear system of first and second differential order, the naming convention should be written as follows

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

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

We recommend to use for nonlinear models

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

An author can use another notation in the case when the convention above is not appropriate.

===Matrix format for linear non-parametric or parametric affine systems===

For the linear non-parametric systems and the parametric affine systems, the matrices of the systems are constant matrices. Such matrices should be written in the Matrix Market format as described at http://math.nist.gov/MatrixMarket/.

A file for a matrix should be named as

problem_name.matrix_name

If there is no file for a matrix, it is assumed to be identity for the E matrix and 0 for the D matrix.

All matrix files for a given problem should be compressed in a single zip or tar.gz archive, there should be a folder in the archive which is named with "problem_name" or with the abbreviated problem name.

===Matrix format for nonlinear or parametric non-affine systems===

For the parametric non-affine systems, or nonlinear systems, the system matrices are coupled with the parameters and/or the state vectors. In this case, the system matrices should be analytically described if possible. For example, <math>A(1,1)=x_1^2+x_3</math>, <math>A(i,2)=\mu_i^3 x_i, i=1,2,\ldots m</math>, etc., where <math>\mu_i</math> are the parameters, and <math>x_i</math> is the <math>i</math>th entry in the state vector <math>x</math>. If the system matrices are too complicated to be described analytically, or in plain text, the source code to generate the matrices should be available (or at
least should be sent after contacting the responsible person).

==References==

* 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.
* 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.
* 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.

Editors:
{{editors}}

Vertical Stand

2022-03-24T10:35:10Z

Mitchell: Changed this again; it seems the slide is moved to discrete positions, not sliding. Still not 100% sure tho!

[[Category:benchmark]]
[[Category:parametric 1 parameter]]
[[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>.



==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]]''

Vertical Stand

2022-03-24T10:32:51Z

Mitchell: Significantly edited a hard-to-understand sentence or two; someone review to ensure that the intended meaning wasn't changed.

[[Category:benchmark]]
[[Category:parametric 1 parameter]]
[[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 moves 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>.



==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]]''

Vertical Stand

2022-03-24T10:28:53Z

Mitchell: Reordered sentence for better readability

[[Category:benchmark]]
[[Category:parametric 1 parameter]]
[[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. Caused by a machining process a tool slide is moving on 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>.



==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]]''

Vertical Stand

2022-03-24T10:25:51Z

Mitchell: typos

[[Category:benchmark]]
[[Category:parametric 1 parameter]]
[[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. On one of its surfaces a pair of guide rails is located. Caused by a machining process a tool slide is moving on 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>.



==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]]''

Vertical Stand

2022-03-24T10:24:07Z

Mitchell: Typo

[[Category:benchmark]]
[[Category:parametric 1 parameter]]
[[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. On one of its surfaces a pair of guide rails is located. Caused by a machining process a tool slide is moving on 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 to be equal. Still, according to the number of layers, for this example, it holds <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>.



==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]]''

Supersonic Engine Inlet

2022-03-15T12:00:15Z

Mitchell: hyphen needed

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:DAE order unspecified]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:MIMO]]
[[Category:Sparse]]

==Description: Active Control of a Supersonic Engine Inlet==

<figure id="fig1">[[File:Supersonic_Inlet1.png|750px|thumb|right|<caption>Steady-state Mach contours inside diffuser. Freestream Mach number
is 2.2.</caption>]]</figure>

This example considers unsteady flow through a [[wikipedia:Diffuser_(thermodynamics)#Supersonic_Diffusers|supersonic diffuser]] as shown in Fig. 1.
The diffuser operates at a nominal [[wikipedia:Mach_number|Mach number]] of <math>2.2</math>, however it is subject to perturbations in the incoming flow, which may be due (for
example) to atmospheric variations.
In nominal operation, there is a strong shock downstream of the diffuser throat, as can be seen from the Mach contours
plotted in Figure Fig. 1.
Incoming disturbances can cause the shock to move forward towards the throat. When the shock sits at the throat, the inlet
is unstable, since any disturbance that moves the shock slightly upstream will cause it to move forward rapidly, leading to unstart of the inlet. This is extremely undesirable, since unstart results in a large loss of thrust.
In order to prevent unstart from occurring, one option is to actively control the position of the shock.
This control may be effected through flow bleeding upstream of the diffuser throat.

A complete description of the benchmark and some model reduction results can be downloaded as PDF file [https://www.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SupersonicEngineInlet.pdf here].

===Active Flow Control Setup===

<figure id="fig2">[[File:Supersonic_Inlet2.png|600px|thumb|right|<caption>Supersonic diffuser active flow control problem setup.</caption>]]</figure>

Fig. 2 presents the schematic of the actuation mechanism.
Incoming flow with possible disturbances enters the inlet and is sensed using pressure sensors.
The controller then adjusts the bleed upstream of the throat in order to control the position of the shock and to prevent it from moving upstream.
In simulations, it is difficult to automatically determine the shock location.
The average Mach number at the diffuser throat provides an appropriate surrogate that can be easily computed.
There are several transfer functions of interest in this problem.
The shock position will be controlled by monitoring the average Mach number at the diffuser throat.
The reduced-order model must capture the dynamics of this output in response to two inputs: the incoming flow disturbance and the bleed actuation.
In addition, total pressure measurements at the diffuser wall are used for sensing.

===CFD Formulation===

The unsteady, two-dimensional flow of an inviscid, compressible fluid is governed by the Euler equations.
The usual statements of mass, momentum, and energy can be written in integral form as

:<math>
\begin{align}
\frac{\partial}{\partial t}\iint\rho\mathrm{d}V + \oint\rho Q\cdot\mathrm{dA} & = 0,\\
\frac{\partial}{\partial t}\iint\rho Q\mathrm{d}V + \oint\rho Q (Q\cdot\mathrm{dA}) + \oint p \mathrm{dA} & = 0,\\
\frac{\partial}{\partial t}\iint\rho E\mathrm{d}V + \oint\rho H (Q\cdot\mathrm{dA}) + \oint p Q\cdot\mathrm{dA} & = 0,
\end{align}
</math>

where <math>\rho</math>, <math>Q</math>, <math>H</math>, <math>E</math>, and <math>p</math> denote density, flow velocity, total enthalpy, energy, and pressure, respectively.
The CFD formulation for this problem uses a finite volume method and is described fully in <ref name="lassaux2002"/> and <ref name="willcox2005"/>.
The unknown flow quantities used are the density, streamwise velocity component, normal velocity component, and enthalpy at each point in the computational grid.
Note that the local flow velocity components <math>q</math> and <math>q^{\perp}</math> are defined using a streamline computational grid that is computed for the steady-state solution.
<math>q</math> is the projection of the flow velocity on the meanline direction of the grid cell, and <math>q^{\perp}</math> is the normal-to-meanline component.
To simplify the implementation of the integral energy equation, total enthalpy is also used in place of energy.
The vector of unknowns at each node <math>i</math> is therefore

:<math>
x_{i} = \begin{bmatrix} \rho_{i}, & q_{i}, & q^{\perp}_{i}, & H_{i} \end{bmatrix}^{T}.
</math>

Two physically different kinds of boundary conditions exist: inflow/outflow conditions, and conditions applied at a solid wall.
At a solid wall, the usual no-slip condition of zero normal flow velocity is easily applied as <math>q^{\perp} = 0</math>.
In addition, we will allow for mass addition or removal (bleed) at various positions along the wall.
The bleed condition is also easily specified.
We set

:<math>
q^{\perp} = \frac{\dot{m}}{\rho},
</math>

where <math>\dot{m}</math> is the specified mass flux per unit length along the bleed slot.
At inflow boundaries, Riemann boundary conditions are used.
For the diffuser problem considered here, all inflow boundaries are supersonic, and hence we impose inlet vorticity, entropy and Riemann’s invariants.
At the exit of the duct, we impose outlet pressure.

===Linearized CFD Matrices===

The two-dimensional integral Euler equations are linearized about the steady-state solution to obtain an unsteady system of the form

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

The descriptor matrix <math>E</math> arises from the particular CFD formulation.
In addition, the matrix <math>E</math> contains some zero rows that are due to implementation of boundary conditions.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38866, see also <ref name="willcox2005"/>.

==Data==

The matrices of this benchmark can be downloaded in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format:
* [http://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/SupersonicEngineInlet-dim1e4-Inlet.tar.gz SupersonicEngineInlet-dim1e4-Inlet.tar.gz].
The size of the file is 5.4 MB.
The matrix name is used as an extension of the matrix file.

==Dimensions==

System structure:

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

System dimensions:

<math>E \in \mathbb{R}^{11730 \times 11730}</math>,
<math>A \in \mathbb{R}^{11730 \times 11730}</math>,
<math>B \in \mathbb{R}^{11730 \times 2}</math>,
<math>C \in \mathbb{R}^{1 \times 11730}</math>

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection, '''Supersonic Engine Inlet'''. hosted at MORwiki - Model Order Reduction Wiki, 2005. http://modelreduction.org/index.php/Supersonic_Engine_Inlet

@MISC{morwiki_supsonengine,
author = <nowiki>{{Oberwolfach Benchmark Collection}}</nowiki>,
title = {Supersonic Engine Inlet},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Supersonic_Engine_Inlet}</nowiki>,
year = 2005
}

* For the background on the benchmark:

@MASTERSTHESIS{morLas13,
author = {G. Lassaux},
year = 2002,
title = {High-Fidelity Reduced-Order Aerodynamic Models: Application to
Active Control of Engine Inlets},
school = {Massachusetts Institute of Technology},
address = {Cambridge, USA},
url = <nowiki>{http://web.mit.edu/kwillcox/Public/Web/LassauxMS.pdf}</nowiki>
}

==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="lassaux2002">G. Lassaux. [http://hdl.handle.net/1721.1/82238 High-Fidelity Reduced-Order Aerodynamic Models: Application to Active Control of Engine Inlets]. Master’s thesis, Dept. of Aeronautics and Astronautics, MIT, June 2002.</ref>

<ref name="willcox2005">K. Willcox , G. Lassaux, [https://doi.org/10.1007/3-540-27909-1_20 Model Reduction of an Actively Controlled Supersonic Diffuser]. In: Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 357--361, 2005.</ref>

</references>

Silicon Nitride Membrane

2022-03-15T11:57:50Z

Mitchell: I believe "optimical" should be "optical" but a bit unclear (optimal?)

[[Category:benchmark]]
[[Category:parametric 2-5 parameters]]
[[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_3)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 2}</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]

Silicon Nitride Membrane

2022-03-15T11:56:28Z

Mitchell: typo

[[Category:benchmark]]
[[Category:parametric 2-5 parameters]]
[[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 optimical 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_3)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 2}</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]

Nonlinear RC Ladder

2022-03-15T11:34:15Z

Mitchell: typo

[[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 ([[List_of_abbreviations#SISO|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]]''

Linear 1D Beam

2022-03-15T10:16:24Z

Mitchell: grammar

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

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

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

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

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

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

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

===Model Description===

This model describes a slender beam with four degrees of freedom per node:

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

See Fig. 2 for Degree of Freedom <math>x</math>, Fig. 3 for Degree of Freedom <math>\theta_x</math> and Fig. 4 for Degrees of freedom <math>y</math> and <math>\theta_z</math>.

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

==Origin==

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

==Data==

Based on the finite element discretization presented in<ref name="Weaver1990"/>, an interactive matrix generator has been created using [http://www.wolfram.com/ Wolfram Research]'s [http://www.wolfram.com/products/webmathematica webMathematica].
However, models produced by this generator are in the <tt>DSIF</tt><ref name="lienemsann2005"/> format, which allows for nonlinear terms.
For the purpose of the benchmark collection, we have precomputed four systems and converted them to the [http://math.nist.gov/MatrixMarket/ Matrix market] format which is easier to import in standard computer algebra packages.

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

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

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

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|File
|Degrees of freedom
|Number of nodes
|Number of equations
|File size [B]
|Compressed size [B]
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Linear1dBeam-dim1e1-LF10.zip Linear1dBeam-dim1e1-LF10.zip]
|flexural (<math>y</math> and <math>\theta_z</math>)
|10
|18
|5935
|2384
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Linear1dBeam-dim1e4-LF10000.zip Linear1dBeam-dim1e4-LF10000.zip]
|flexural (<math>y</math> and <math>\theta_z</math>)
|10000
|19998
|6640324
|716807
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Linear1dBeam-dim1e1-LFAT5.zip Linear1dBeam-dim1e1-LFAT5.zip]
|flexural (<math>y</math> and <math>\theta_z</math>), axial, torsional
|5
|14
|4045
|2255
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Linear1dBeam-dim1e5-LFAT5000.zip Linear1dBeam-dim1e5-LFAT5000.zip]
|flexural (<math>y</math> and <math>\theta_z</math>), axial, torsional
|50000
|19994
|5532532
|627991
|}

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

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

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

Details of the implementation are available in a separate [https://www.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Linear1dBeam.pdf report]. A typical input to this system is a step response; periodic on/off switching is also possible.
The reduced model should thus both represent the step response as well as the possible influence of higher order harmonics.
See also <ref name="lienemann2006"/>.

==Dimensions==

System structure:

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

System dimensions:

<math>M \in \mathbb{R}^{N \times N}</math>,
<math>E \in \mathbb{R}^{N \times N}</math>,
<math>K \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times N}</math>

System variants:

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

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

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

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

* For the background on the benchmark:

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

==References==

<references>

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

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

<ref name="Weaver1990"> W. Weaver Jr., S.P. Timoshenko, D.H. Young, [https://books.google.de/books?id=YZ7t8LgRqi0C&lpg=PP1&dq=editions%3AgXRPzjrsO3IC&pg=PP1#v=onepage&q&f=false Vibration problems in engineering], 5th ed., Wiley, 1990.</ref>

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

</references>

Linear 1D Beam

2022-03-15T10:15:35Z

Mitchell: wrong word

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

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

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

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

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

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

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

===Model Description===

This model describes a slender beam with four degrees of freedom per node:

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

See Fig. 2 for Degree of Freedom <math>x</math>, Fig. 3 for Degree of Freedom <math>\theta_x</math> and Fig. 4 for Degrees of freedom <math>y</math> and <math>\theta_z</math>.

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

==Origin==

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

==Data==

Based on the finite element discretization presented in<ref name="Weaver1990"/>, an interactive matrix generator has been created using [http://www.wolfram.com/ Wolfram Research]'s [http://www.wolfram.com/products/webmathematica webMathematica].
However, models produced by this generator are in the <tt>DSIF</tt><ref name="lienemsann2005"/> format, which allows for nonlinear terms.
For the purpose of the benchmark collection, we have precomputed four systems and converted them to the [http://math.nist.gov/MatrixMarket/ Matrix market] format which is easier to import in standard computer algebra packages.

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

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

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

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|File
|Degrees of freedom
|Number of nodes
|Number of equations
|File size [B]
|Compressed size [B]
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Linear1dBeam-dim1e1-LF10.zip Linear1dBeam-dim1e1-LF10.zip]
|flexural (<math>y</math> and <math>\theta_z</math>)
|10
|18
|5935
|2384
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Linear1dBeam-dim1e4-LF10000.zip Linear1dBeam-dim1e4-LF10000.zip]
|flexural (<math>y</math> and <math>\theta_z</math>)
|10000
|19998
|6640324
|716807
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Linear1dBeam-dim1e1-LFAT5.zip Linear1dBeam-dim1e1-LFAT5.zip]
|flexural (<math>y</math> and <math>\theta_z</math>), axial, torsional
|5
|14
|4045
|2255
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Linear1dBeam-dim1e5-LFAT5000.zip Linear1dBeam-dim1e5-LFAT5000.zip]
|flexural (<math>y</math> and <math>\theta_z</math>), axial, torsional
|50000
|19994
|5532532
|627991
|}

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

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

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

Details of the implementation are available in a separate [https://www.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Linear1dBeam.pdf report]. A typical input to this system is a step response; periodic on/off switching is also possible.
The reduced model should thus both represent the step response as well as the possible influence of higher order harmonics.
See also <ref name="lienemann2006"/>.

==Dimensions==

System structure:

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

System dimensions:

<math>M \in \mathbb{R}^{N \times N}</math>,
<math>E \in \mathbb{R}^{N \times N}</math>,
<math>K \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times N}</math>

System variants:

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

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

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

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

* For the background on the benchmark:

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

==References==

<references>

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

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

<ref name="Weaver1990"> W. Weaver Jr., S.P. Timoshenko, D.H. Young, [https://books.google.de/books?id=YZ7t8LgRqi0C&lpg=PP1&dq=editions%3AgXRPzjrsO3IC&pg=PP1#v=onepage&q&f=false Vibration problems in engineering], 5th ed., Wiley, 1990.</ref>

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

</references>

Inverse Lyapunov Procedure

2022-03-15T10:13:42Z

Mitchell: typo

[[Category:benchmark]]
[[Category:procedural]]
[[Category:linear]]
[[Category:first differential order]]
[[Category:time invariant]]
[[Category:MIMO]]

==Description==
The '''Inverse Lyapunov Procedure''' (ILP) is a synthetic random linear system generator.
It is based on reversing the [[Balanced_Truncation|Balanced Truncation]] procedure and was developed in <ref name="smith03"/>,
where a description of the algorithm is given.
In aggregate form, for randomly generated controllability and observability gramians, a balancing transformation is computed.
The balanced gramian is the basis for an associated state-space system,
which is determined by solving a [[wikipedia:Lyapunov_equation|Lyapunov equation]] and then unbalanced.
A central point is the solution of the Lyapunov equations for the system matrix instead of the gramian matrix.
This is feasable due to the symmetric (semi-)positive definiteness of the gramians and the requirement for a stable system, yet with a non-unique solution.
Following, the steps of the ILP are listed:

# Sample eigenvalues of controllability and observability Gramians.
# Generate random orthogonal matrices (ie.: SVD of random matrix).
# Compute balancing transformation for these Gramians.
# Sample random input and output matrices.
# Scale output matrix to input matrix.
# Solve Lyapunov equation for system matrix.
# Unbalance system.

===Inverse Sylvester Procedure===
A variant of the '''Inverse Lyapunov Procedure''' is the inverse Sylvester procedure (ILS) <ref name="himpe17"/>,
which generates only state-space symmetric systems.
Instead of balanced truncation, the [[wikipedia:Cross_Gramian|cross Gramian]] is utilized for the random system generation, and hence a [[wikipedia:Sylvester_equation|Sylvester equation]] is needs to be solved.
The steps for the ILS are listed below:

# Sample cross Gramian eigenvalues.
# Sample random input matrix, and set output matrix as its transpose.
# Solve Sylvester equation for system matrix.
# Sample orthogonal unbalancing transformation (QR of random matrix).
# Unbalance system.

Even though the ILS is more limited than the ILP, for large systems it can be more efficient.

==Data==
This benchmark is procedural and the input, state and output dimensions can be chosen.
Use the following [http://matlab.com MATLAB] code to generate a random system as described above:

<div class="thumbinner" style="width:20%;text-align:left;">
<source lang="matlab">

function [A,B,C] = ilp(M,N,Q,s,r)
% ilp (inverse lyapunov procedure)
% by Christian Himpe, 2013--2018
% released under BSD 2-Clause License
%*
if(nargin==5)
rand('seed',r);
randn('seed',r);
end;

% Gramian Eigenvalues
WC = exp(0.5*rand(N,1));
WO = exp(0.5*rand(N,1));

% Gramian Eigenvectors
[P,S,R] = svd(randn(N));

% Balancing Transformation
WC = P*diag(sqrt(WC))*P';
WO = R*diag(sqrt(WO))*R';
[U,D,V] = svd(WC*WO);

% Input and Output
B = randn(N,M);

if(nargin>=4 && s~=0)
C = B';
else
C = randn(Q,N);
end

% Scale Output Matrix
BB = sum(B.*B,2); % = diag(B*B')
CC = sum(C.*C,1)'; % = diag(C'*C)
C = bsxfun(@times,C,sqrt(BB./CC)');

% Solve System Matrix
A = -sylvester(D,D,B*B');

% Unbalance System
A = V*A*U';
B = V*B;
C = C*U';

end
</source>
</div>

The function call requires three parameters; the number of inputs <math>M</math>, of states <math>N</math> and outputs <math>Q</math>.
Optionally, a symmetric system can be enforced with the parameter <math>s \neq 0</math>.
For reproducibility, the random number generator seed can be controlled by the parameter <math>r \in \mathbb{N}</math>.
The return value consists of three matrices; the system matrix <math>A</math>, the input matrix <math>B</math> and the output matrix <math>C</math>.

:<source lang="matlab">
[A,B,C] = ilp(M,N,Q,s,r);
</source>

A variant of the above code using empirical Gramians instead of a matrix equation solution can be found at http://gramian.de/utils/ilp.m , which may yield preferable results.

==Dimensions==

System structure:
:<math>
\begin{align}
\dot{x}(t) &= Ax(t) + Bu(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 M}</math>,
<math>C \in \mathbb{R}^{Q \times N}</math>.

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

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

@MISC{morwiki-invlyapproc,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Inverse Lyapunov Procedure},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Inverse_Lyapunov_Procedure}</nowiki>,
year = 2018
}

* For the background on the benchmark:

@INPROCEEDINGS{SmiF03,
author = {Smith, S.~C. and Fisher, J.},
title = {On generating random systems: a gramian approach},
booktitle = {Proc. Am. Control. Conf.},
volume = 3,
pages = {2743--2748},
year = 2003,
doi = {10.1109/ACC.2003.1243494}
}

==References==
<references>

<ref name="smith03">S.C. Smith, J. Fisher, "[https://doi.org/10.1109/ACC.2003.1243494 On generating random systems: a gramian approach]", Proceedings of the American Control Conference, 3: 2743--2748, 2003.</ref>

<ref name="himpe17">C. Himpe, M. Ohlberger, "[https://doi.org/10.1007/978-3-319-58786-8_17 Cross-Gramian-Based Model Reduction: A Comparison]", In: Model Reduction of Parametrized Systems, Modeling, Simulation and Applications, vol. 17: 271--283, 2017.</ref>

</references>

==Contact==
[[User:Himpe|Christian Himpe]]

Hydro-Electric Open Channel

2022-03-15T10:11:41Z

Mitchell: typos

[[Category:benchmark]]

==Description==

===Motivation===

The so-called [[wikipedia:Shallow_water_equations|Saint-Venant equations]] are largely used in the hydraulic domain to model the dynamics of an open channel flow. These equations consist of two nonlinear hyperbolic PDEs. In the considered benchmark, under mild simplifying assumptions detailed in <ref name= Dalmas2016>V. Dalmas, G. Robert, C. Poussot-Vassal, I. Pontes-Duff and C. Seren, "[https://doi.org/10.1109/ECC.2016.7810582 From infinite dimensional modelling to parametric reduced order approximation: Application to open-channel flow for hydroelectricity]", in Proceedings of the European Control Conference (ECC), Aalborg, Denmark, July, 2016, pp. 1982-1987.</ref>, the St Venant PDE equations describing the height variation <math>h</math> of the river as a function of the inflow <math>q_i</math> and outflow <math>q_o</math> variations, at location <math>x</math> (<math>x\in[0\,\,L]</math>, <math>L\in\mathbb R_+</math>), obtained around some flow and height linearisation point, can be formulated as follows:
:<math>
h(x,s) = \mathbf{G_i}(x,s)q_i(s) - \mathbf{G_o}(x,s)q_o(s) = \mathbf H(x,s) u(s).
</math>

The <math>\mathbf G_i</math> and <math>\mathbf G_o</math> are the following irrational functions:
:<math>
\mathbf{G_i}(x,s)= \dfrac{\lambda_1(s)e^{\lambda_2(s)L+\lambda_1(s)x}-\lambda_2(s)e^{\lambda_1(s)L+\lambda_2(s)x}}{B_0s(e^{\lambda_1(s)L}-e^{\lambda_2(s)L})}
</math>
and
:<math>
\mathbf{G_o}(x,s)= \dfrac{\lambda_1(s)e^{\lambda_1(s)x}-\lambda_2(s)e^{\lambda_2(s)x}}{B_0s(e^{\lambda_1(s)L}-e^{\lambda_2(s)L})}
</math>

===Considered data===

The benchmark contains the above irrational model description together with the numerical data as used in the reference paper, given as Matlab handle functions.

==Origin==

Collaboration between [https://www.onera.fr ONERA] and [https://www.edf.fr/ EDF]. The data from the simplified hydro-electric open-channel model come from V. Dalmas and G. Robert while and the post-processing was performed jointly by P. Vuillemin, and C. Poussot-Vassal.

==Data==

===Description===

The [[File:HydroelectricChannel.zip]] (67KB) repository contains three files:

* The <tt>dataONERA_Hydroelectric.mat</tt> data file, with the data describing the irrational model described above

* The <tt>dataONERA_Hydroelectric_withMOR.mat</tt> data file, with 2 ROMs obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the Loewner method
** Hr1 : linear rational ROM (state-space models in Matlab form).
** Hr2 : linear rational ROM with post stability enforcement and addition of the 0 singularity afterward (state-space models in Matlab form).

* The <tt>startONERA_Hydroelectric.m</tt> script file, used to load and plot the data for illustration.

===Objective===

The model's main dynamics are a delayed one with an integral action.
Find a (linear) reduced order model stable but with one single singularity in 0 that approximates the irrational model over some frequency range (e.g. [0,0.01]rad/s)

===Remark===
As the original model is irrational and of infinite dimension, it exhibits an infinite number of singularities. Therefore, approximation over the complete frequency range is unachievable, at least with a rational function. That is why an approximation over a bounded frequency range is preferable. Indeed, strange behaviors may appear.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Hydro-Electric Open Channel'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Hydro-Electric_Open_Channel

* For the background on the benchmark:

@inproceedings{DalmasECC:2016,
author = {V. Dalmas and G. Robert and C. Poussot-Vassal and I. {Pontes Duff} and C. Seren},
title = {From infinite dimensional modelling to parametric reduced order approximation: Application to open-channel flow for hydroelectricity},
booktitle = {Proceedings of the 15th European Control Conference},
year = {2016},
pages = {1982--1987},
doi = {10.1109/ECC.2016.7810582}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Hydro-Electric Open Channel

2022-03-15T10:10:12Z

Mitchell: fixed a sentence

[[Category:benchmark]]

==Description==

===Motivation===

The so-called [[wikipedia:Shallow_water_equations|Saint-Venant equations]] are largely used in the hydraulic domain to model the dynamics of an open channel flow. These equations consist of two nonlinear hyperbolic PDEs. In the considered benchmark, under mild simplifying assumptions detailed in <ref name= Dalmas2016>V. Dalmas, G. Robert, C. Poussot-Vassal, I. Pontes-Duff and C. Seren, "[https://doi.org/10.1109/ECC.2016.7810582 From infinite dimensional modelling to parametric reduced order approximation: Application to open-channel flow for hydroelectricity]", in Proceedings of the European Control Conference (ECC), Aalborg, Denmark, July, 2016, pp. 1982-1987.</ref>, the St Venant PDE equations describing the height variation <math>h</math> of the river as a function of the inflow <math>q_i</math> and outflow <math>q_o</math> variations, at location <math>x</math> (<math>x\in[0\,\,L]</math>, <math>L\in\mathbb R_+</math>), obtained around some flow and height linearisation point, can be formulated as follows:
:<math>
h(x,s) = \mathbf{G_i}(x,s)q_i(s) - \mathbf{G_o}(x,s)q_o(s) = \mathbf H(x,s) u(s).
</math>

The <math>\mathbf G_i</math> and <math>\mathbf G_o</math> are the following irrational functions:
:<math>
\mathbf{G_i}(x,s)= \dfrac{\lambda_1(s)e^{\lambda_2(s)L+\lambda_1(s)x}-\lambda_2(s)e^{\lambda_1(s)L+\lambda_2(s)x}}{B_0s(e^{\lambda_1(s)L}-e^{\lambda_2(s)L})}
</math>
and
:<math>
\mathbf{G_o}(x,s)= \dfrac{\lambda_1(s)e^{\lambda_1(s)x}-\lambda_2(s)e^{\lambda_2(s)x}}{B_0s(e^{\lambda_1(s)L}-e^{\lambda_2(s)L})}
</math>

===Considered data===

The benchmark contains the above irrational model description together with the numerical data as used in the reference paper, given as Matlab handle functions.

==Origin==

Collaboration between [https://www.onera.fr ONERA] and [https://www.edf.fr/ EDF]. The data from the simplified hydro-electric open-channel model come from V. Dalmas and G. Robert while and the post-processing was performed jointly by P. Vuillemin, and C. Poussot-Vassal.

==Data==

===Description===

The [[File:HydroelectricChannel.zip]] (67KB) repository contains three files:

* The <tt>dataONERA_Hydroelectric.mat</tt> data file, with the data describing the irrational model described above

* The <tt>dataONERA_Hydroelectric_withMOR.mat</tt> data file, with 2 ROMs obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the Loewner method
** Hr1 : linear rational ROM (state-space models in Matlab form).
** Hr2 : linear rational ROM with post stability enforcement and addition of the 0 singularity afterward (state-space models in Matlab form).

* The <tt>startONERA_Hydroelectric.m</tt> script file, used to load and plot the data for illustration.

===Objective===

The model's main dynamics are a delayed one with an integral action.
Find a (linear) reduced order model stable but with one single singularity in 0 that approximates the irrational model over some frequency range (e.g. [0,0.01]rad/s)

===Remark===
As the original model is irrational and of infinite dimension, it exhibits an infinite number of singularities. Therefore, approximation over the complete frequency range is un-achievable, at least with a rational function. That is why an approximation over a bounded frequency range is prefferable. Indeed, strange behaviours may appear.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Hydro-Electric Open Channel'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Hydro-Electric_Open_Channel

* For the background on the benchmark:

@inproceedings{DalmasECC:2016,
author = {V. Dalmas and G. Robert and C. Poussot-Vassal and I. {Pontes Duff} and C. Seren},
title = {From infinite dimensional modelling to parametric reduced order approximation: Application to open-channel flow for hydroelectricity},
booktitle = {Proceedings of the 15th European Control Conference},
year = {2016},
pages = {1982--1987},
doi = {10.1109/ECC.2016.7810582}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Gas Sensor

2022-03-15T10:06:23Z

Mitchell: wrong word

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

==Description: Microhotplate Gas Sensor==
<figure id="fig1">[[File:GasSensor1.jpg|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:GasSensor2.jpg|490px|thumb|right|<caption>Masks disposition (left) and the schematical position of the chosen output nodes (right).</caption>]]</figure>

The goal of European project [https://web.archive.org/web/20050826075758/http://www.cnm.es:80/imb/glassgas/index.htm Glassgas] (IST-99-19003) was to develop a novel metal oxide low power '''microhotplate gas sensor''' <ref name="wollenstein03"/>.
In order to assure a robust design and good thermal isolation of the membrane from the surrounding wafer, the silicon microhotplate is supported by glass pillars emanating from a glass cap above the silicon wafer, as shown in Fig. 1.
In this design, four different sensitive layers can be deposited on the membrane.
The thermal management of a '''microhotplate gas sensor''' is of crucial importance.

The benchmark contains a thermal model of a single gas sensor device with three main components:
a silicon rim, a silicon hotplate and glass structure <ref name="hildenbrand03"/>.
It allows us to simulate important thermal issues, such as the homogeneous temperature distribution over gas sensitive regions or thermal decoupling between the hotplate and the silicon rim.
The original model is the heat transfer partial differential equation.

The device solid model has been made and then meshed and discretized in [http://www.ansys.com ANSYS] 6.1 by means of the finite element method (<tt>SOLID70</tt> elements were used).
It contains 68000 elements and 73955 nodes.
Material properties were considered as temperature independent.
Temperature is assumed to be in degree Celsius with the initial state of <math>0 C</math>.
The Dirichlet boundary conditions of <math>T = 0 C</math> is applied at the top and bottom of the chip (at 7038 nodes).

The output nodes are described in Table 1.
In Fig. 2 the red marked nodes are positioned on the silicon rim.
Their temperature should be close to the initial temperature in the case of good thermal decoupling between the membrane and the silicon rim.
The black marked nodes are placed on the sensitive layers above the heater and are numbered from left to right row by row, as schematically shown in Fig 2.
They allow us to prove whether the temperature distribution over the gas sensitive layers is homogeneous (maximum difference of <math>10C</math> is allowed by design).

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Inputs and outputs for the gas sensor model.''
|-
|Number
|Code
|Comment
|-
|1
|aHeater
|within a heater, to be used for nonlinear input
|-
|2-7
|SiRim1 to SiRim7
|silicon rim
|-
|8-28
|Memb1 to Memb21
|gas sensitive layer
|}

The benchmark contains a constant load vector.
The input function equal to <math>u(t) = 1</math> corresponds to the constant input power of <math>340 mW</math>.
One can insert a weak input nonlinearity related to the dependence of heater's resistivity on temperature given as:

:<math>
R(T) = R_{0}(1 + \alpha T)
</math>

where <math>\alpha =1.469 \cdot 10^{-3} K^{-1}</math>.
To this end, one has to multiply the load vector by a function:

:<math>
\frac{U^2 274.94 (1 + \alpha T)}{0.34 (274.94 (1 + \alpha T)+148.13)^2}
</math>

where <math>U</math> is a desired constant voltage.
The temperature in the equation above should be replaced by the temperature at the input 1 (aHeater).

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

:<math>
\begin{array}{rcl}
E \frac{\partial T}{\partial t} &=& A T(t) + B u(t) \\
y(t) &=& C T(t)
\end{array}
</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, and <math>T</math> is the vector of unknown temperatures.
The dimension of the system is <math>66917</math>, the number of nonzero elements in matrix <math>E</math> is <math>66917</math>, in matrix <math>A</math> is <math>885141</math>.

The outputs of the transient simulation at output 18 (Memb11) over the rise time of the device of <math>5 s</math> for the original linear (with constant input power of <math>340 mW</math>) and nonlinear (with constant voltage of <math>14 V</math>) model are placed in files <tt>LinearResults</tt> and <tt>NonlinearResults</tt> respectively.
The results can be used to compare the solution of a reduced model with the original one.
The time integration has been performed in ANSYS with accuracy of about <math>0.1 \%</math>.
The results are given as matrices where the first row is made of times, the second of the temperatures.

More information can also be found in <ref name="bechthold05"/>.

==Data==

Download matrices in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format: [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/GasSensor-dim1e5-GasSensor.tar.gz GasSensor-dim1e5-GasSensor.tar.gz], (8 MB).

The matrix name is used as an extension of the matrix file.
File <tt>*.C</tt> names contains a list of output names written consecutively.
The system matrices have been extracted from ANSYS models by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem].

The discussion of electro-thermal modeling related to the benchmark including the nonlinear input function can be found in <ref name="bechthold04"/>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38880, see <ref name="hildenbrand2005"/>.

==Dimensions==

System structure:

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

System dimensions:

<math>E \in \mathbb{R}^{66917 \times 66917}</math>,
<math>A \in \mathbb{R}^{66917 \times 66917}</math>,
<math>B \in \mathbb{R}^{66917 \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times 66917}</math>.

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

@INPROCEEDINGS{BecHWetal04,
author = <nowiki>{T. Bechtold, J. Hildenbrand, J. Wöllenstein, J. G. Korvink}</nowiki>,
title = {Model Order Reduction of 3D Electro-Thermal Model for a Novel, Micromachined Hotplate Gas Sensor},
booktitle = {Proceedings of 5th International Conference on Thermal and Mechanical Simulation and Experiments in Microelectronics and Microsystems},
pages = {263--267},
year = {2004},
doi = {10.1109/ESIME.2004.1304049}
}

==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="hildenbrand2005">Hildenbrand J., Bechtold T., J. Wöllenstein, [https://doi.org/10.1007/3-540-27909-1_14 Microhotplate Gas Sensor]. In: Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol 45: 333-336, 2005.</ref>

<ref name="wollenstein03">J. Wöllenstein, H. Böttner, J.A. Pláza, C. Carné, Y. Min, H.L. Tuller, [https://doi.org/10.1016/S0925-4005(03)00218-1 A novel single chip thin film metal oxide array], Sensors and Actuators B: Chemical 93 (1-3): 350--355, 2003.</ref>

<ref name="hildenbrand03">J. Hildenbrand, [https://www.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/GasSensor-Hildenbrand03.pdf Simulation and Characterisation of a Gas sensor and Preparation for Model Order Reduction], Diploma Thesis, University of Freiburg, Germany, 2003.</ref>

<ref name="bechthold04">T. Bechtold, J. Hildenbrand, J. Wöllenstein, J. G. Korvink, [https://doi.org/10.1109/ESIME.2004.1304049 Model Order Reduction of 3D Electro-Thermal Model for a Novel, Micromachined Hotplate Gas Sensor], Proceedings of 5th International Conference on Thermal and Mechanical Simulation and Experiments in Microelectronics and Microsystems, EUROSIME2004, May 10-12, 2004, Brussels, Belgium: 263--267, 2004.</ref>

<ref name="bechthold05">T. Bechtold, "[https://www.freidok.uni-freiburg.de/volltexte/1914/ Model Order Reduction of Electro-Thermal MEMS]", PhD thesis, Department of Microsystems Engineering, University of Freiburg, 2005.</ref>

</references>

==Contact==

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

[https://www.jade-hs.de/team/tamara-bechtold/ Tamara Bechtold]

Electrostatic Beam

2022-03-14T15:54:06Z

Mitchell: typo

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

==Description: Beam Actuated by Electrostatic Force==

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

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

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

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

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

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

===Model description===

This model describes a slender beam which is actuated by a voltage between the beam and the ground electrode below (see Fig. 1).

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

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

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

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

==Origin==

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

==Data==

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

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

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
! Property
! Value
|-
|Beam length (l)
|<math>10^{-5} m</math>
|-
|Beam height (h)
|<math>10^{-6} m</math>
|-
|Beam width (w)
|<math>15 \cdot 10^{-6} m</math>
|-
|Distance between beams (s)
|<math>200 \cdot 10^{-9} m</math>
|-
|Material density (rho)
|<math>2330 kg/m^3</math>
|-
|Cross-sectional area (A)
|<math>150 \cdot 10^{12} m^2</math>
|-
|Moment of inertia (I)
|<math>1.25 \cdot 10^{-21} m^4</math>
|-
|Modulus of elasticity (E)
|<math>1.31 \cdot 10^{11} Pa</math>
|-
|Contribution of <math>M</math> to damping
|<math>0</math>
|-
|Contribution of <math>K</math> to damping
|<math>10^{-6}</math>
|-
|Support
|Both sides, <math>y</math> DOF only
|}

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

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
! File
! Number of Nodes
! Number of Equations
! Compressed Size [kB]
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ElectrostaticBeam-dim1e1-E10.zip ElectrostaticBeam-dim1e1-E10.zip]
|10
|38
|4144
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ElectrostaticBeam-dim1e2-E100.zip ElectrostaticBeam-dim1e2-E100.zip]
|100
|398
|347679
|}

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

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

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

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

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

==Dimensions==

System structure:

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

System dimensions:

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

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

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

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

==References==

<references>

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

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

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

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

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

</references>

Electrostatic Beam

2022-03-14T15:52:11Z

Mitchell: grammar

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

==Description: Beam Actuated by Electrostatic Force==

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

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

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

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

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

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

===Model description===

This model describes a slender beam which is actuated by a voltage between the beam and the ground electrode below (see Fig. 1).

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

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

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

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

==Origin==

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

==Data==

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

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

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
! Property
! Value
|-
|Beam length (l)
|<math>10^{-5} m</math>
|-
|Beam height (h)
|<math>10^{-6} m</math>
|-
|Beam width (w)
|<math>15 \cdot 10^{-6} m</math>
|-
|Distance between beams (s)
|<math>200 \cdot 10^{-9} m</math>
|-
|Material density (rho)
|<math>2330 kg/m^3</math>
|-
|Cross-sectional area (A)
|<math>150 \cdot 10^{12} m^2</math>
|-
|Moment of inertia (I)
|<math>1.25 \cdot 10^{-21} m^4</math>
|-
|Modulus of elasticity (E)
|<math>1.31 \cdot 10^{11} Pa</math>
|-
|Contribution of <math>M</math> to damping
|<math>0</math>
|-
|Contribution of <math>K</math> to damping
|<math>10^{-6}</math>
|-
|Support
|Both sides, <math>y</math> DOF only
|}

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

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
! File
! Number of Nodes
! Number of Equations
! Compressed Size [kB]
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ElectrostaticBeam-dim1e1-E10.zip ElectrostaticBeam-dim1e1-E10.zip]
|10
|38
|4144
|-
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/ElectrostaticBeam-dim1e2-E100.zip ElectrostaticBeam-dim1e2-E100.zip]
|100
|398
|347679
|}

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

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

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

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

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

==Dimensions==

System structure:

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

System dimensions:

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

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

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

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

==References==

<references>

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

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

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

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

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

</references>

Coplanar Waveguide

2022-03-14T15:14:59Z

Mitchell: errant comma

[[Category:benchmark]]
[[Category:PDE]]
[[Category:parametric 2-5 parameters]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:affine parameter representation]]

==Description==

A '''coplanar waveguide''' (see Fig. 1) is a microwave semiconductor device, which is governed by [[wikipedia:Maxwell's_equations|Maxwell's equations]].
The [[wikipedia:Coplanar_waveguide|coplanar waveguide]] considered with dielectric overlay, i.e. a transmission line shielded within two layers of multilayer board with <math>0.5mm</math> thickness are buried in a substrate with <math>10mm</math> thickness and relative permittivity
<math>\epsilon_r = 4.4 </math> and relative permeability <math>\mu_r = 1 </math>, and low conductivity <math>\sigma = 0.02 S/m </math>.
The low-loss upper layer has low permittivity <math>\epsilon_r = 1.07 </math> and <math>\sigma = 0.01 S/m</math>.
The whole structure is enlosed in a metallic box of dimension <math>140mm</math> by <math>100mm</math> by <math>50mm</math>.
The discrete port with <math>50Ohm</math> lumped load imposes <math>1 A</math> current as the input to the one side of the strip.
The voltage along the discrete port 2 at the end of the other side of coupled lines is integrated as the output.

<figure id="fig:coplanar">
[[File:CoplanarWaveguideScaled.jpg|frame|<caption>Coplanar Waveguide Model<ref name="hess2003"/></caption>]]
</figure>

==Data==

Considered parameters are the frequency <math> \omega </math> and the width <math> \nu </math> of the middle stripline.

The affine form <math> a(u, v; \omega, \nu) = \sum_{q=1}^Q \Theta^q(\omega, \nu) a^q(u, v) </math> can be established using <math> Q = 15 </math> affine terms.

The discretized bilinear form is <math> a(u, v; \omega, \nu) = \sum_{q=1}^Q \Theta^q(\omega, \nu) 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 H(curl) inner product matrix have been assembled
using the Finite Element Method, resulting in 7754 degrees of freedom, after removal of boundary conditions. The files are numbered according to their
appearance in the summation.

The coefficient functions are given by:

:<math> \Theta^1(\omega, \nu) = 1 </math>

:<math> \Theta^2(\omega, \nu) = \omega </math>

:<math> \Theta^3(\omega, \nu) = -\omega^2 </math>

:<math> \Theta^4(\omega, \nu) = \frac{\nu}{6} </math>

:<math> \Theta^5(\omega, \nu) = \frac{6}{\nu} </math>

:<math> \Theta^6(\omega, \nu) = \frac{6 \omega}{\nu} </math>

:<math> \Theta^7(\omega, \nu) = -\frac{6 \omega^2}{\nu} </math>

:<math> \Theta^8(\omega, \nu) = \frac{\nu \omega}{6} </math>

:<math> \Theta^9(\omega, \nu) = -\frac{\nu \omega^2}{6} </math>

:<math> \Theta^{10}(\omega, \nu) = \frac{16 - \nu}{10} </math>

:<math> \Theta^{11}(\omega, \nu) = \frac{10}{16 - \nu} </math>

:<math> \Theta^{12}(\omega, \nu) = \frac{10 \omega}{16 - \nu} </math>

:<math> \Theta^{13}(\omega, \nu) = -\frac{10 \omega^2}{16 - \nu} </math>

:<math> \Theta^{14}(\omega, \nu) = \frac{16 - \nu}{10} \omega </math>

:<math> \Theta^{15}(\omega, \nu) = -\frac{16 - \nu}{10} \omega^2 </math>

The parameter domain of interest is <math>\omega \in [0.6, 3.0] \cdot 10^9 </math> Hz, where the factor of <math>10^9 </math> has already been taken into account
while assembling the matrices, while the geometric variation occurs between <math> \nu \in [2.0, 14.0]</math>.
The input functional also has a factor of <math> \omega </math>.

There are two output functionals, which is due to the fact that the complex system has been rewritten as a real symmetric one.
In particular the computation of the output

<math>s(u) = | l^T * u |</math> with complex vector <math>u</math> turns into <math>s(u) = \sqrt{ (l_1^T * u)^2 + (l_2^T * u)^2 }</math> with real vector <math> u </math>.

==Origin==

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

==Data==

The files are numbered according to their appearance in the summation and can be found here: [[Media:Matrices_cp.tar.gz|Matrices_cp.tar.gz]].

==Dimensions==

System structure:

:<math>
\begin{array}{rcl}
\sum_{q=1}^{15} \Theta^q(\omega, \nu) A^q u &=& b \\
s &=& \sqrt{ (l_1^T * u)^2 + (l_2^T * u)^2 }
\end{array}
</math>

System dimensions:

<math>A^q \in \mathbb{R}^{15504 \times 15504}</math>,
<math>b \in \mathbb{R}^{15504}</math>,
<math>l_1 \in \mathbb{R}^{15504}</math>,
<math>l_2 \in \mathbb{R}^{15504}</math>.

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

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

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

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

==References==

<references>

<ref name="hess2003">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]]''

Convective Thermal Flow

2022-03-14T15:12:16Z

Mitchell: typo

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

==Description: Convective Thermal Flow Problems==
<figure id="fig1">[[File:Convection1.jpg|490px|thumb|right|Figure 1: Convective heat flow example: 2D anemometer model]]</figure>
<figure id="fig2">[[File:Convection2.jpg|490px|thumb|right|Figure 2: Convective heat flow example: 3D cooling structure]]</figure>

Many thermal problems require simulation of heat exchange between a solid body and a fluid flow.
The most elaborate approach to this problem is [[wikipedia:Computational_fluid_dynamics|computational fluid dynamics]] (CFD).
However, CFD is computationally expensive.
A popular solution is to exclude the flow completely from the computational domain and to use convection boundary conditions for the solid model.
However, caution has to be taken to select the [[wikipedia:Heat_transfer_coefficient|film coefficient]].

An intermediate level is to include a flow region with a given velocity profile that adds convective transport to the model.
Compared to convection boundary conditions this approach has the advantage that the film coefficient does not need to be specified and that information about the heat profile in the flow can be obtained.
A drawback of the method is the greatly increased number of elements needed to perform a physically valid simulation because the solution accuracy when employing upwind finite element schemes depends on the element size.
While this problem still is linear, due to the forced convection, the conductivity matrix changes from a symmetric matrix to an un-symmetric one.
So this problem type can be used as a benchmark for problems containing un-symmetric matrices.

Two different designs are tested: a 2D model of an [[Anemometer|anemometer]]-like structure mainly consisting of a tube and a small heat source (Fig. 1) <ref name="ernst2001"/>.
The solid model has been generated and meshed in [http://www.ansys.com ANSYS].
Triangular <tt>PLANE55</tt> elements have been used for meshing and discretizing by the finite element method, resulting in 19282 elements and 9710 nodes.
The second design is a 3D model of a chip cooled by forced convection (Fig. 2) <ref name="harper1997"/>.
In this case the tetrahedral element type <tt>SOLID70</tt> was used, resulting in 107989 elements and 20542 nodes.
Since the implementation of the convective term in ANSYS does not allow defining the fluid speed on a per element basis but on a per region one, the flow profile has to be approximated by piece-wise step functions.
The approximation used for these benchmarks is shown in Fig. 1.

The Dirichlet boundary conditions are applied to the original system.
In both models the reference temperature is set to <math>300 K</math>, Dirichlet boundary conditions as well as initial conditions are set to <math>0</math> with respect to the reference.
The specified Dirichlet boundary conditions are in both cases the inlet of the fluid and the outer faces of the solids. Matrices are supplied for the symmetric case (fluid speed is zero; no convection), and the non-symmetric case (with forced convection).
Table 1 shows the output nodes specified for the two benchmarks, Table 2 links the filenames according to the different cases.

Practically, only a few nodes are considered quantities of interest.
Hence, a small subset of five nodes is selected as output nodes,
which are filtered from the discretized state by a linear transformation.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38867, see <ref name="moosmann2005"/>.

==Data==

Matrices are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format.
The matrix name is used as an extension of the matrix file.
<tt>*.C.names</tt> contains a list of output names written consecutively.
The system matrices have been extracted from ANSYS models by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem].

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Output nodes for the two models.''
|-
!Model
!Number
!Code
!Comment
|-
|Flow Meter
|1
|out1
|outlet position
|-
|
|2
|out2
|outlet position
|-
|
|3
|SenL
|left sensor position
|-
|
|4
|Heater
|within the heater
|-
|
|5
|SenR
|right sensor position
|-
|Cooling Structure
|1
|out1
|outlet position
|-
|
|2
|out2
|outlet position
|-
|
|3
|out3
|outlet position
|-
|
|4
|out4
|outlet position
|-
|
|5
|Heater
|within the heater
|}

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 2: Provided files.''
|-
!Model
!Fluid Speed (m/s)
!Link
!Size
|-
|Flow Meter
|<math>0</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-flow_meter_model_v0.tgz Convection-dim1e4-flow_meter_model_v0.tgz]
|649.4 kB
|-
|
|<math>0.5</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-flow_meter_model_v0.5.tgz Convection-dim1e4-flow_meter_model_v0.5.tgz]
|757.8 kB
|-
|Cooling Structure
|<math>0</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-chip_cooling_model_v0.tgz Convection-dim1e4-chip_cooling_model_v0.tgz]
|3.9 MB
|-
|
|<math>0.1</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-chip_cooling_model_v0.1.tgz Convection-dim1e4-chip_cooling_model_v0.1.tgz]
|4.0 MB
|}

Further information on the models can be found in <ref name="moosmann2004"/>,
where model reduction by means of the [[wikipedia:Arnoldi_iteration|Arnoldi algorithm]] is also presented.

==Dimensions==

System structure:

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

System dimensions:

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

System variants:

<tt>flow_v0</tt>: <math>N = 9669</math>,
<tt>flow_v0.5</tt>: <math>N = 9669</math>,
<tt>chip_v0</tt>: <math>N = 20082</math>,
<tt>chip_v0.1</tt>: <math>N = 20082</math>.

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

@INPROCEEDINGS{morMooRGetal04,
author = <nowiki>{C. Moosmann, E.B. Rudnyi, A. Greiner, J.G. Korvink}</nowiki>,
title = {Model Order Reduction for Linear Convective Thermal Flow},
booktitle = {Proceedings of 10th International Workshops on THERMal INvestigations of ICs and Systems},
year = {2004},
url = <nowiki>{http://modelreduction.com/doc/papers/moosmann04THERMINIC.pdf}</nowiki>
}

==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="ernst2001">H. Ernst, [https://freidok.uni-freiburg.de/data/201 High-Resolution Thermal Measurements in Fluids], PhD thesis, University of Freiburg, Germany, 2001.</ref>

<ref name="harper1997">C.A. Harper, [https://doi.org/10.1036/0071430482 Electronic packaging and interconnection handbook], New York McGraw- Hill, USA, 1997</ref>

<ref name="moosmann2004">C. Moosmann, E.B. Rudnyi, A. Greiner, 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, Sophia Antipolis, France, 2004.</ref>

<ref name="moosmann2005">C. Moosmann, A. Greiner, [https://doi.org/10.1007/3-540-27909-1_16 Convective Thermal Flow Problems], In: Dimension Reduction of Large-Scale Systems. Springer, Berlin, Heidelberg. Lecture Notes in Computational Science and Engineering, vol 45: 341--343, 2005.</ref>

</references>

Convective Thermal Flow

2022-03-14T15:10:52Z

Mitchell: grammar

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

==Description: Convective Thermal Flow Problems==
<figure id="fig1">[[File:Convection1.jpg|490px|thumb|right|Figure 1: Convective heat flow example: 2D anemometer model]]</figure>
<figure id="fig2">[[File:Convection2.jpg|490px|thumb|right|Figure 2: Convective heat flow example: 3D cooling structure]]</figure>

Many thermal problems require simulation of heat exchange between a solid body and a fluid flow.
The most elaborate approach to this problem is [[wikipedia:Computational_fluid_dynamics|computational fluid dynamics]] (CFD).
However, CFD is computationally expensive.
A popular solution is to exclude the flow completely from the computational domain and to use convection boundary conditions for the solid model.
However, caution has to be taken to select the [[wikipedia:Heat_transfer_coefficient|film coefficient]].

An intermediate level is to include a flow region with a given velocity profile that adds convective transport to the model.
Compared to convection boundary conditions this approach has the advantage that the film coefficient does not need to be specified and that information about the heat profile in the flow can be obtained.
A drawback of the method is the greatly increased number of elements needed to perform a physically valid simulation because the solution accuracy when employing upwind finite element schemes depends on the element size.
While this problem still is linear, due to the forced convection, the conductivity matrix changes from a symmetric matrix to an un-symmetric one.
So this problem type can be used as a benchmark for problems containing un-symmetric matrices.

Two different designs are tested: a 2D model of an [[Anemometer|anemometer]]-like structure mainly consisting of a tube and a small heat source (Fig. 1) <ref name="ernst2001"/>.
The solid model has been generated and meshed in [http://www.ansys.com ANSYS].
Triangular <tt>PLANE55</tt> elements have been used for meshing and discretizing by the finite element method, resulting in 19282 elements and 9710 nodes.
The second design is a 3D model of a chip cooled by forced convection (Fig. 2) <ref name="harper1997"/>.
In this case the tetrahedral element type <tt>SOLID70</tt> was used, resulting in 107989 elements and 20542 nodes.
Since the implementation of the convective term in ANSYS does not allow defining the fluid speed on a per element basis but on a per region one, the flow profile has to be approximated by piece-wise step functions.
The approximation used for this benchmarks is shown in Fig. 1.

The Dirichlet boundary conditions are applied to the original system.
In both models the reference temperature is set to <math>300 K</math>, Dirichlet boundary conditions as well as initial conditions are set to <math>0</math> with respect to the reference.
The specified Dirichlet boundary conditions are in both cases the inlet of the fluid and the outer faces of the solids. Matrices are supplied for the symmetric case (fluid speed is zero; no convection), and the non-symmetric case (with forced convection).
Table 1 shows the output nodes specified for the two benchmarks, Table 2 links the filenames according to the different cases.

Practically, only a few nodes are considered quantities of interest.
Hence, a small subset of five nodes is selected as output nodes,
which are filtered from the discretized state by a linear transformation.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38867, see <ref name="moosmann2005"/>.

==Data==

Matrices are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format.
The matrix name is used as an extension of the matrix file.
<tt>*.C.names</tt> contains a list of output names written consecutively.
The system matrices have been extracted from ANSYS models by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem].

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Output nodes for the two models.''
|-
!Model
!Number
!Code
!Comment
|-
|Flow Meter
|1
|out1
|outlet position
|-
|
|2
|out2
|outlet position
|-
|
|3
|SenL
|left sensor position
|-
|
|4
|Heater
|within the heater
|-
|
|5
|SenR
|right sensor position
|-
|Cooling Structure
|1
|out1
|outlet position
|-
|
|2
|out2
|outlet position
|-
|
|3
|out3
|outlet position
|-
|
|4
|out4
|outlet position
|-
|
|5
|Heater
|within the heater
|}

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 2: Provided files.''
|-
!Model
!Fluid Speed (m/s)
!Link
!Size
|-
|Flow Meter
|<math>0</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-flow_meter_model_v0.tgz Convection-dim1e4-flow_meter_model_v0.tgz]
|649.4 kB
|-
|
|<math>0.5</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-flow_meter_model_v0.5.tgz Convection-dim1e4-flow_meter_model_v0.5.tgz]
|757.8 kB
|-
|Cooling Structure
|<math>0</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-chip_cooling_model_v0.tgz Convection-dim1e4-chip_cooling_model_v0.tgz]
|3.9 MB
|-
|
|<math>0.1</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-chip_cooling_model_v0.1.tgz Convection-dim1e4-chip_cooling_model_v0.1.tgz]
|4.0 MB
|}

Further information on the models can be found in <ref name="moosmann2004"/>,
where model reduction by means of the [[wikipedia:Arnoldi_iteration|Arnoldi algorithm]] is also presented.

==Dimensions==

System structure:

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

System dimensions:

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

System variants:

<tt>flow_v0</tt>: <math>N = 9669</math>,
<tt>flow_v0.5</tt>: <math>N = 9669</math>,
<tt>chip_v0</tt>: <math>N = 20082</math>,
<tt>chip_v0.1</tt>: <math>N = 20082</math>.

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

@INPROCEEDINGS{morMooRGetal04,
author = <nowiki>{C. Moosmann, E.B. Rudnyi, A. Greiner, J.G. Korvink}</nowiki>,
title = {Model Order Reduction for Linear Convective Thermal Flow},
booktitle = {Proceedings of 10th International Workshops on THERMal INvestigations of ICs and Systems},
year = {2004},
url = <nowiki>{http://modelreduction.com/doc/papers/moosmann04THERMINIC.pdf}</nowiki>
}

==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="ernst2001">H. Ernst, [https://freidok.uni-freiburg.de/data/201 High-Resolution Thermal Measurements in Fluids], PhD thesis, University of Freiburg, Germany, 2001.</ref>

<ref name="harper1997">C.A. Harper, [https://doi.org/10.1036/0071430482 Electronic packaging and interconnection handbook], New York McGraw- Hill, USA, 1997</ref>

<ref name="moosmann2004">C. Moosmann, E.B. Rudnyi, A. Greiner, 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, Sophia Antipolis, France, 2004.</ref>

<ref name="moosmann2005">C. Moosmann, A. Greiner, [https://doi.org/10.1007/3-540-27909-1_16 Convective Thermal Flow Problems], In: Dimension Reduction of Large-Scale Systems. Springer, Berlin, Heidelberg. Lecture Notes in Computational Science and Engineering, vol 45: 341--343, 2005.</ref>

</references>

Convective Thermal Flow

2022-03-14T15:09:05Z

Mitchell: errant comma

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

==Description: Convective Thermal Flow Problems==
<figure id="fig1">[[File:Convection1.jpg|490px|thumb|right|Figure 1: Convective heat flow example: 2D anemometer model]]</figure>
<figure id="fig2">[[File:Convection2.jpg|490px|thumb|right|Figure 2: Convective heat flow example: 3D cooling structure]]</figure>

Many thermal problems require simulation of heat exchange between a solid body and a fluid flow.
The most elaborate approach to this problem is [[wikipedia:Computational_fluid_dynamics|computational fluid dynamics]] (CFD).
However, CFD is computationally expensive.
A popular solution is to exclude the flow completely from the computational domain and to use convection boundary conditions for the solid model.
However, caution has to be taken to select the [[wikipedia:Heat_transfer_coefficient|film coefficient]].

An intermediate level is to include a flow region with a given velocity profile that adds convective transport to the model.
Compared to convection boundary conditions this approach has the advantage that the film coefficient does not need to be specified and that information about the heat profile in the flow can be obtained.
A drawback of the method is the greatly increased number of elements needed to perform a physically valid simulation because the solution accuracy when employing upwind finite element schemes depends on the element size.
While this problem still is linear, due to the forced convection, the conductivity matrix changes from a symmetric matrix to an un-symmetric one.
So this problem type can be used as a benchmark for problems containing un-symmetric matrices.

Two different designs are tested: a 2D model of an [[Anemometer|anemometer]]-like structure mainly consisting of a tube and a small heat source (Fig. 1) <ref name="ernst2001"/>.
The solid model has been generated and meshed in [http://www.ansys.com ANSYS].
Triangular <tt>PLANE55</tt> elements have been used for meshing and discretizing by the finite element method, resulting in 19282 elements and 9710 nodes.
The second design is a 3D model of a chip cooled by forced convection (Fig. 2) <ref name="harper1997"/>.
In this case the tetrahedral element type <tt>SOLID70</tt> was used, resulting in 107989 elements and 20542 nodes.
Since the implementation of the convective term in ANSYS does not allow for definition of the fluid speed on a per element, but on a per region basis, the flow profile has to be approximated by piece-wise step functions.
The approximation used for this benchmarks is shown in Fig. 1.

The Dirichlet boundary conditions are applied to the original system.
In both models the reference temperature is set to <math>300 K</math>, Dirichlet boundary conditions as well as initial conditions are set to <math>0</math> with respect to the reference.
The specified Dirichlet boundary conditions are in both cases the inlet of the fluid and the outer faces of the solids. Matrices are supplied for the symmetric case (fluid speed is zero; no convection), and the non-symmetric case (with forced convection).
Table 1 shows the output nodes specified for the two benchmarks, Table 2 links the filenames according to the different cases.

Practically, only a few nodes are considered quantities of interest.
Hence, a small subset of five nodes is selected as output nodes,
which are filtered from the discretized state by a linear transformation.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38867, see <ref name="moosmann2005"/>.

==Data==

Matrices are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format.
The matrix name is used as an extension of the matrix file.
<tt>*.C.names</tt> contains a list of output names written consecutively.
The system matrices have been extracted from ANSYS models by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem].

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Output nodes for the two models.''
|-
!Model
!Number
!Code
!Comment
|-
|Flow Meter
|1
|out1
|outlet position
|-
|
|2
|out2
|outlet position
|-
|
|3
|SenL
|left sensor position
|-
|
|4
|Heater
|within the heater
|-
|
|5
|SenR
|right sensor position
|-
|Cooling Structure
|1
|out1
|outlet position
|-
|
|2
|out2
|outlet position
|-
|
|3
|out3
|outlet position
|-
|
|4
|out4
|outlet position
|-
|
|5
|Heater
|within the heater
|}

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 2: Provided files.''
|-
!Model
!Fluid Speed (m/s)
!Link
!Size
|-
|Flow Meter
|<math>0</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-flow_meter_model_v0.tgz Convection-dim1e4-flow_meter_model_v0.tgz]
|649.4 kB
|-
|
|<math>0.5</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-flow_meter_model_v0.5.tgz Convection-dim1e4-flow_meter_model_v0.5.tgz]
|757.8 kB
|-
|Cooling Structure
|<math>0</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-chip_cooling_model_v0.tgz Convection-dim1e4-chip_cooling_model_v0.tgz]
|3.9 MB
|-
|
|<math>0.1</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-chip_cooling_model_v0.1.tgz Convection-dim1e4-chip_cooling_model_v0.1.tgz]
|4.0 MB
|}

Further information on the models can be found in <ref name="moosmann2004"/>,
where model reduction by means of the [[wikipedia:Arnoldi_iteration|Arnoldi algorithm]] is also presented.

==Dimensions==

System structure:

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

System dimensions:

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

System variants:

<tt>flow_v0</tt>: <math>N = 9669</math>,
<tt>flow_v0.5</tt>: <math>N = 9669</math>,
<tt>chip_v0</tt>: <math>N = 20082</math>,
<tt>chip_v0.1</tt>: <math>N = 20082</math>.

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

@INPROCEEDINGS{morMooRGetal04,
author = <nowiki>{C. Moosmann, E.B. Rudnyi, A. Greiner, J.G. Korvink}</nowiki>,
title = {Model Order Reduction for Linear Convective Thermal Flow},
booktitle = {Proceedings of 10th International Workshops on THERMal INvestigations of ICs and Systems},
year = {2004},
url = <nowiki>{http://modelreduction.com/doc/papers/moosmann04THERMINIC.pdf}</nowiki>
}

==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="ernst2001">H. Ernst, [https://freidok.uni-freiburg.de/data/201 High-Resolution Thermal Measurements in Fluids], PhD thesis, University of Freiburg, Germany, 2001.</ref>

<ref name="harper1997">C.A. Harper, [https://doi.org/10.1036/0071430482 Electronic packaging and interconnection handbook], New York McGraw- Hill, USA, 1997</ref>

<ref name="moosmann2004">C. Moosmann, E.B. Rudnyi, A. Greiner, 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, Sophia Antipolis, France, 2004.</ref>

<ref name="moosmann2005">C. Moosmann, A. Greiner, [https://doi.org/10.1007/3-540-27909-1_16 Convective Thermal Flow Problems], In: Dimension Reduction of Large-Scale Systems. Springer, Berlin, Heidelberg. Lecture Notes in Computational Science and Engineering, vol 45: 341--343, 2005.</ref>

</references>

Convective Thermal Flow

2022-03-14T15:08:14Z

Mitchell: grammar

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

==Description: Convective Thermal Flow Problems==
<figure id="fig1">[[File:Convection1.jpg|490px|thumb|right|Figure 1: Convective heat flow example: 2D anemometer model]]</figure>
<figure id="fig2">[[File:Convection2.jpg|490px|thumb|right|Figure 2: Convective heat flow example: 3D cooling structure]]</figure>

Many thermal problems require simulation of heat exchange between a solid body and a fluid flow.
The most elaborate approach to this problem is [[wikipedia:Computational_fluid_dynamics|computational fluid dynamics]] (CFD).
However, CFD is computationally expensive.
A popular solution is to exclude the flow completely from the computational domain and to use convection boundary conditions for the solid model.
However, caution has to be taken to select the [[wikipedia:Heat_transfer_coefficient|film coefficient]].

An intermediate level is to include a flow region with a given velocity profile that adds convective transport to the model.
Compared to convection boundary conditions this approach has the advantage that the film coefficient does not need to be specified and that information about the heat profile in the flow can be obtained.
A drawback of the method is the greatly increased number of elements needed to perform a physically valid simulation, because the solution accuracy when employing upwind finite element schemes depends on the element size.
While this problem still is linear, due to the forced convection, the conductivity matrix changes from a symmetric matrix to an un-symmetric one.
So this problem type can be used as a benchmark for problems containing un-symmetric matrices.

Two different designs are tested: a 2D model of an [[Anemometer|anemometer]]-like structure mainly consisting of a tube and a small heat source (Fig. 1) <ref name="ernst2001"/>.
The solid model has been generated and meshed in [http://www.ansys.com ANSYS].
Triangular <tt>PLANE55</tt> elements have been used for meshing and discretizing by the finite element method, resulting in 19282 elements and 9710 nodes.
The second design is a 3D model of a chip cooled by forced convection (Fig. 2) <ref name="harper1997"/>.
In this case the tetrahedral element type <tt>SOLID70</tt> was used, resulting in 107989 elements and 20542 nodes.
Since the implementation of the convective term in ANSYS does not allow for definition of the fluid speed on a per element, but on a per region basis, the flow profile has to be approximated by piece-wise step functions.
The approximation used for this benchmarks is shown in Fig. 1.

The Dirichlet boundary conditions are applied to the original system.
In both models the reference temperature is set to <math>300 K</math>, Dirichlet boundary conditions as well as initial conditions are set to <math>0</math> with respect to the reference.
The specified Dirichlet boundary conditions are in both cases the inlet of the fluid and the outer faces of the solids. Matrices are supplied for the symmetric case (fluid speed is zero; no convection), and the non-symmetric case (with forced convection).
Table 1 shows the output nodes specified for the two benchmarks, Table 2 links the filenames according to the different cases.

Practically, only a few nodes are considered quantities of interest.
Hence, a small subset of five nodes is selected as output nodes,
which are filtered from the discretized state by a linear transformation.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38867, see <ref name="moosmann2005"/>.

==Data==

Matrices are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format.
The matrix name is used as an extension of the matrix file.
<tt>*.C.names</tt> contains a list of output names written consecutively.
The system matrices have been extracted from ANSYS models by means of [http://portal.uni-freiburg.de/imteksimulation/downloads/mor4fem mor4fem].

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 1: Output nodes for the two models.''
|-
!Model
!Number
!Code
!Comment
|-
|Flow Meter
|1
|out1
|outlet position
|-
|
|2
|out2
|outlet position
|-
|
|3
|SenL
|left sensor position
|-
|
|4
|Heater
|within the heater
|-
|
|5
|SenR
|right sensor position
|-
|Cooling Structure
|1
|out1
|outlet position
|-
|
|2
|out2
|outlet position
|-
|
|3
|out3
|outlet position
|-
|
|4
|out4
|outlet position
|-
|
|5
|Heater
|within the heater
|}

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|''Table 2: Provided files.''
|-
!Model
!Fluid Speed (m/s)
!Link
!Size
|-
|Flow Meter
|<math>0</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-flow_meter_model_v0.tgz Convection-dim1e4-flow_meter_model_v0.tgz]
|649.4 kB
|-
|
|<math>0.5</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-flow_meter_model_v0.5.tgz Convection-dim1e4-flow_meter_model_v0.5.tgz]
|757.8 kB
|-
|Cooling Structure
|<math>0</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-chip_cooling_model_v0.tgz Convection-dim1e4-chip_cooling_model_v0.tgz]
|3.9 MB
|-
|
|<math>0.1</math>
|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/Convection-dim1e4-chip_cooling_model_v0.1.tgz Convection-dim1e4-chip_cooling_model_v0.1.tgz]
|4.0 MB
|}

Further information on the models can be found in <ref name="moosmann2004"/>,
where model reduction by means of the [[wikipedia:Arnoldi_iteration|Arnoldi algorithm]] is also presented.

==Dimensions==

System structure:

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

System dimensions:

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

System variants:

<tt>flow_v0</tt>: <math>N = 9669</math>,
<tt>flow_v0.5</tt>: <math>N = 9669</math>,
<tt>chip_v0</tt>: <math>N = 20082</math>,
<tt>chip_v0.1</tt>: <math>N = 20082</math>.

==Citation==

To cite this benchmark, use the following references:

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

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

* For the background on the benchmark:

@INPROCEEDINGS{morMooRGetal04,
author = <nowiki>{C. Moosmann, E.B. Rudnyi, A. Greiner, J.G. Korvink}</nowiki>,
title = {Model Order Reduction for Linear Convective Thermal Flow},
booktitle = {Proceedings of 10th International Workshops on THERMal INvestigations of ICs and Systems},
year = {2004},
url = <nowiki>{http://modelreduction.com/doc/papers/moosmann04THERMINIC.pdf}</nowiki>
}

==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="ernst2001">H. Ernst, [https://freidok.uni-freiburg.de/data/201 High-Resolution Thermal Measurements in Fluids], PhD thesis, University of Freiburg, Germany, 2001.</ref>

<ref name="harper1997">C.A. Harper, [https://doi.org/10.1036/0071430482 Electronic packaging and interconnection handbook], New York McGraw- Hill, USA, 1997</ref>

<ref name="moosmann2004">C. Moosmann, E.B. Rudnyi, A. Greiner, 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, Sophia Antipolis, France, 2004.</ref>

<ref name="moosmann2005">C. Moosmann, A. Greiner, [https://doi.org/10.1007/3-540-27909-1_16 Convective Thermal Flow Problems], In: Dimension Reduction of Large-Scale Systems. Springer, Berlin, Heidelberg. Lecture Notes in Computational Science and Engineering, vol 45: 341--343, 2005.</ref>

</references>

Artificial Fishtail

2022-03-14T13:13:41Z

Mitchell: typo

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

<figure id="fig:plot1">
[[File:Fishtail.png|4380px|thumb|right|<caption>Schematic 3D-Model-Fishtail</caption>]]
</figure>
<figure id="fig:plot2">
[[File:fishtail_tf.png|438px|thumb|right|<caption>Sigma Plot Fishtail Cp</caption>]]
</figure>
<figure id="fig:plot3">
[[File:fishtail_ext_tf.png|438px|thumb|right|<caption>Sigma Plot Fishtail Cp_ext</caption>]]
</figure>

==Description==
Todays [[wikipedia:Autonomous_underwater_vehicle|autonomous underwater vehicles]] (AUVs) are a source of noise pollution and inefficiency due to their screw propeller driven design.
The evolution of fish has, on the other hand, optimized their underwater efficiency and agility over millennia.
The adaption of fish-like drive systems for AUVs is therefore a promising choice.

==Model Description==
This model describes the silicon body of an artificial fishtail supported by a central carbon beam.
The rear part of the fish-body without the fins is modeled as as a 3D FEM model using linear elasticity.
In the current stage of modeling the tail is rigidly mounted in the front, the states in <math>x</math> represent the displacements of the finite element degrees of freedom.
The fish-like locomotion is enabled by pumping air between two sets of pressure chambers in the left and right halves of the tail.
The single input <math>u</math> of the system is thus the pumping pressure.
The outputs are displacements of certain surface points.
There are two variants of the model.
The first has three outputs representing the displacements of the point of interest, the rear tip of the carbon beam, in the three spatial directions.
For the second variant six additional points <math>(z_1,z_2,z_3)</math> on the flank are added as outputs, yielding a total of 21 outputs.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
! scope="col" style="width: 12ex;" | <math>z_1</math>
! scope="col" style="width: 12ex;" | <math>z_2</math>
! scope="col" style="width: 12ex;" | <math>z_3</math>
|-
| 0.05
| 0.0
| 0.0
|-
| 0.0474526
| 0.0
| 0.0599584
|-
| 0.04032111
| 0.0
| 0.105274
|-
| 0.0326229
| 0.0
| 0.136726
|-
| 0.0250675
| 0.0
| 0.16107
|-
| 0.0168069
| 0.0
| 0.183588
|-
| 0.0
| 0.0
| 0.21
|}

Note that the POI (Point of Interest) is the last row in this table and in Cp_ext in the data files (see below).
The additional outputs show two effects.
On the one hand, for purely input output related reduction methods they avoid drastic deviations on the interior states,
on the other hand they show a smoothing effect for the models transfer function.

==Origin==
The model was set up and computed at the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Artificial_Fishtail chair of automatic control] at CAU Kiel and first presented in <ref name="SieKM18" />.

==Data==
The model is based on the the finite element package [https://www.firedrakeproject.org Firedrake] and uses the material parameters:
{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Part
|Parameter
|Value
|Unit
|-style="background-color:#FFFFFF;"
|
|<math>\varrho_1</math>
|<math>1.07\cdot 10^{−3}</math>
|<math>\frac{\text{kg}}{\text{m}^{3}}</math>
|-style="background-color:#FFFFFF;"
|Hull
|<math>E_1</math>
|<math>0.025 \cdot 10^6</math>
|<math>\frac{\text{kg}}{\text{m}\text{s}^2}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\nu_1</math>
|<math>0.48</math>
|
|-
|
|<math>\varrho_2</math>
|<math>1.4 \cdot 10^{3}</math>
|<math>\frac{\text{kg}}{\text{m}^{3}}</math>
|-
| Beam
|<math> E_2</math>
|<math> 2.96 \cdot 10^{10}</math>
|<math> \frac{\text{kg}}{\text{m}\text{s}^2}</math>
|-
|
|<math>\nu_2</math>
|<math> 0.3</math>
|
|-style="background-color:#FFFFFF;"
| Rayleigh damping
|<math>\alpha_r</math>
|<math> 1.0 \cdot 10^{-4}</math>
|<math>\frac{1}{s}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\beta_r</math>
|<math> 2.0 \cdot 10^{-4}</math>
|<math>\text{s}</math>
|}

==Dimensions==
System structure:

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

System dimensions:

<math>M \in \mathbb{R}^{N \times N}</math>,
<math>E \in \mathbb{R}^{N \times N}</math>,
<math>K \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times M}</math>,
<math>C \in \mathbb{R}^{P \times N}</math>,
with <math>N=779\,232</math> and <math>M=1</math>.

The internal damping is modeled as Rayleigh damping <math>E=\alpha_r M + \beta_r K</math> using the coefficients from the table above.

System variants:

* <math>P = 3</math>: <math>C=</math><tt>Cp</tt> in the [https://doi.org/10.5281/zenodo.2558728 data files],
* <math>P = 21</math>: <math>C=</math><tt>Cp_ext</tt> in the [https://doi.org/10.5281/zenodo.2558728 data files].

== Remarks ==
* Physically meaningful inputs are of dimension <math>u(t) = \mathcal{O}(10^3)</math>. As an example, a step signal with around <math>5\,000</math>Pa leads to a horizontal POI displacement of about <math>3</math>cm.
* The interesting operation frequencies are in the range between <math>0</math>Hz and <math>10</math>Hz.
* If required, the finite element mesh behind the model and a CSV file with the output locations are available [https://doi.org/10.5281/zenodo.2565173 separately].

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

* For the benchmark itself and its data:
@Misc{SieKM19,
author = {Siebelts, D. and Kater, A. and Meurer, T. and Andrej, J.},
title = {Matrices for an Artificial Fishtail},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2019,
doi = {10.5281/zenodo.2558728}
}

* For the background on the benchmark:

@Article{SieKM18,
author = {Siebelts, D. and Kater, A. and Meurer, T.},
title = {Modeling and Motion Planning for an Artificial Fishtail},
journal = {IFAC-PapersOnLine},
volume = 51,
number = 2,
year = 2018,
pages = {319--324},
doi = {10.1016/j.ifacol.2018.03.055},
publisher = {Elsevier {BV}}
}

==References==
<references>
<ref name="SieKM18">D. Siebelts, A. Kater, T. Meurer, [https://doi.org/doi:10.1016/j.ifacol.2018.03.055 Modeling and Motion Planning for an Artificial Fishtail], IFAC-PapersOnLine (9th Vienna International Conference on Mathematical Modelling) 51(2): 319--324, 2018.</ref>
</references>

Artificial Fishtail

2022-03-14T13:11:16Z

Mitchell: typo

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

<figure id="fig:plot1">
[[File:Fishtail.png|4380px|thumb|right|<caption>Schematic 3D-Model-Fishtail</caption>]]
</figure>
<figure id="fig:plot2">
[[File:fishtail_tf.png|438px|thumb|right|<caption>Sigma Plot Fishtail Cp</caption>]]
</figure>
<figure id="fig:plot3">
[[File:fishtail_ext_tf.png|438px|thumb|right|<caption>Sigma Plot Fishtail Cp_ext</caption>]]
</figure>

==Description==
Todays [[wikipedia:Autonomous_underwater_vehicle|autonomous underwater vehicles]] (AUVs) are a source of noise pollution and inefficiency due to their screw propeller driven design.
The evolution of fish has, on the other hand, optimized their underwater efficiency and agility over millennia.
The adaption of fish-like drive systems for AUVs is therefore a promising choice.

==Model Description==
This model describes the silicon body of an artificial fishtail supported by a central carbon beam.
The rear part of the fish-body without the fins is modeled as as a 3D FEM model using linear elasticity.
In the current stage of modeling the tail is rigidly mounted in the front, the states in <math>x</math> represent the displacements of the finite element degrees of freedom.
The fish-like locomotion is enabled by pumping air between two sets of pressure chambers in the left and right halves of the tail.
The single input <math>u</math> of the system is thus the pumping pressure.
The outputs are displacements of certain surface points.
There are two variants of the model.
The first has three outputs representing the displacements of the point of interest, the rear tip of the carbon beam, in the three spatial directions.
For the second variant six additional points <math>(z_1,z_2,z_3)</math> on the flank are added as outputs, yielding a total of 21 outputs.

{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
! scope="col" style="width: 12ex;" | <math>z_1</math>
! scope="col" style="width: 12ex;" | <math>z_2</math>
! scope="col" style="width: 12ex;" | <math>z_3</math>
|-
| 0.05
| 0.0
| 0.0
|-
| 0.0474526
| 0.0
| 0.0599584
|-
| 0.04032111
| 0.0
| 0.105274
|-
| 0.0326229
| 0.0
| 0.136726
|-
| 0.0250675
| 0.0
| 0.16107
|-
| 0.0168069
| 0.0
| 0.183588
|-
| 0.0
| 0.0
| 0.21
|}

Note that the POI (Point of Interest) is the last row in this table and in Cp_ext in the data files (see below).
The additional outputs show two effects.
On the one hand, for purely input output related reduction methods they avoid drastic deviations on the interior states,
on the other hand they show a smoothing effect for the models transfer function.

==Origin==
The model was setup and computed at the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Artificial_Fishtail chair of automatic control] at CAU Kiel and first presented in <ref name="SieKM18" />.

==Data==
The model is based on the the finite element package [https://www.firedrakeproject.org Firedrake] and uses the material parameters:
{| class="wikitable" style="margin: auto;"
|+ style="caption-side:bottom;"|
|-
|Part
|Parameter
|Value
|Unit
|-style="background-color:#FFFFFF;"
|
|<math>\varrho_1</math>
|<math>1.07\cdot 10^{−3}</math>
|<math>\frac{\text{kg}}{\text{m}^{3}}</math>
|-style="background-color:#FFFFFF;"
|Hull
|<math>E_1</math>
|<math>0.025 \cdot 10^6</math>
|<math>\frac{\text{kg}}{\text{m}\text{s}^2}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\nu_1</math>
|<math>0.48</math>
|
|-
|
|<math>\varrho_2</math>
|<math>1.4 \cdot 10^{3}</math>
|<math>\frac{\text{kg}}{\text{m}^{3}}</math>
|-
| Beam
|<math> E_2</math>
|<math> 2.96 \cdot 10^{10}</math>
|<math> \frac{\text{kg}}{\text{m}\text{s}^2}</math>
|-
|
|<math>\nu_2</math>
|<math> 0.3</math>
|
|-style="background-color:#FFFFFF;"
| Rayleigh damping
|<math>\alpha_r</math>
|<math> 1.0 \cdot 10^{-4}</math>
|<math>\frac{1}{s}</math>
|-style="background-color:#FFFFFF;"
|
|<math>\beta_r</math>
|<math> 2.0 \cdot 10^{-4}</math>
|<math>\text{s}</math>
|}

==Dimensions==
System structure:

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

System dimensions:

<math>M \in \mathbb{R}^{N \times N}</math>,
<math>E \in \mathbb{R}^{N \times N}</math>,
<math>K \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times M}</math>,
<math>C \in \mathbb{R}^{P \times N}</math>,
with <math>N=779\,232</math> and <math>M=1</math>.

The internal damping is modeled as Rayleigh damping <math>E=\alpha_r M + \beta_r K</math> using the coefficients from the table above.

System variants:

* <math>P = 3</math>: <math>C=</math><tt>Cp</tt> in the [https://doi.org/10.5281/zenodo.2558728 data files],
* <math>P = 21</math>: <math>C=</math><tt>Cp_ext</tt> in the [https://doi.org/10.5281/zenodo.2558728 data files].

== Remarks ==
* Physically meaningful inputs are of dimension <math>u(t) = \mathcal{O}(10^3)</math>. As an example, a step signal with around <math>5\,000</math>Pa leads to a horizontal POI displacement of about <math>3</math>cm.
* The interesting operation frequencies are in the range between <math>0</math>Hz and <math>10</math>Hz.
* If required, the finite element mesh behind the model and a CSV file with the output locations are available [https://doi.org/10.5281/zenodo.2565173 separately].

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

* For the benchmark itself and its data:
@Misc{SieKM19,
author = {Siebelts, D. and Kater, A. and Meurer, T. and Andrej, J.},
title = {Matrices for an Artificial Fishtail},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
year = 2019,
doi = {10.5281/zenodo.2558728}
}

* For the background on the benchmark:

@Article{SieKM18,
author = {Siebelts, D. and Kater, A. and Meurer, T.},
title = {Modeling and Motion Planning for an Artificial Fishtail},
journal = {IFAC-PapersOnLine},
volume = 51,
number = 2,
year = 2018,
pages = {319--324},
doi = {10.1016/j.ifacol.2018.03.055},
publisher = {Elsevier {BV}}
}

==References==
<references>
<ref name="SieKM18">D. Siebelts, A. Kater, T. Meurer, [https://doi.org/doi:10.1016/j.ifacol.2018.03.055 Modeling and Motion Planning for an Artificial Fishtail], IFAC-PapersOnLine (9th Vienna International Conference on Mathematical Modelling) 51(2): 319--324, 2018.</ref>
</references>

All pass system

2022-03-14T12:59:30Z

Mitchell: typo

{{preliminary}} 

[[Category:benchmark]]
[[Category:procedural]]
[[Category:linear]]
[[Category:first differential order]]
[[Category:time invariant]]
[[Category:SISO]]

==Description==

This procedural benchmark generates an all-pass SISO system based on <ref name="Obe87"/>.
For an all-pass system, the transfer function has the property <math>g(s)g(-s) = \sigma^2</math>, <math>\sigma > 0</math>,
or (equivalently) the controllability and observability Gramians are quasi inverse to each other: <math>W_C W_O = \sigma I</math>,
which means this system has a singular Hankel singular value of multiplicity of the system's order.
The system matrices are constructing based on the scheme:

:<math>
\begin{align}
A &= \begin{pmatrix} a_{1,1} & -\alpha_1 \\
\alpha_1 & 0 & -\alpha_2 \\
& \ddots & \ddots & \ddots \\
& & \alpha_{N-1} & 0 & -\alpha_{N-1} \end{pmatrix}, \\
B &= \begin{pmatrix} b_1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \\
C &= \begin{pmatrix} s_1 b_1 & 0 & \dots & 0 \end{pmatrix}, \\
D &= -s_1 \sigma.
\end{align}
</math>

We choose <math>s_1 \in \{-1,1\}</math>, to be <math>s_1 \equiv -1</math>, as this makes the system state-space-anti-symmetric.
Furthermore, <math>b_1 = 1</math> and <math>\sigma_1 = 1</math>, which makes <math>a_{1,1} = -\frac{b_1^2}{2 \sigma} = -\frac{1}{2}</math>.

==Data==

This benchmark is procedural and the state dimensions can be chosen.
Use the following [http://matlab.com MATLAB] code to generate a random system as described above:



<div class="thumbinner" style="width:20%;text-align:left;">
<source lang="matlab">

function [A,B,C,D] = allpass(N)
% allpass (all-pass system)
% by Christian Himpe, 2020
% released under BSD 2-Clause License
%*

A = gallery('tridiag',N,-1,0,1);
A(1,1) = -0.5;
B = sparse(1,1,1,N,1);
C = -B';
D = 1;
end
</source>
</div>

The function call requires one argument; the number of states <math>N</math>.
The return value consists of four matrices; the system matrix <math>A</math>, the input matrix <math>B</math>, the output matrix <math>C</math>, and the feed-through matrix <math>D</math>.

:<source lang="matlab">
[A,B,C,D] = allpass(N);
</source>

==Dimensions==

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

System dimensions:

<math>A \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times N}</math>,
<math>D \in \mathbb{R}</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''All-Pass System'''. MORwiki - Model Order Reduction Wiki, 2020. http://modelreduction.org/index.php/All_pass_system

@MISC{morwiki_allpass,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {All-Pass System},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/All_pass_system}</nowiki>,
year = {2020}
}

==References==

<references>

<ref name="Obe87">R.J. Ober. "[https://doi.org/10.1016/S1474-6670(17)55030-2 Asymptotically Stable All-Pass Transfer Functions: Canonical Form, Parametrization and Realization]", IFAC Proceedings Volumes, 20(5): 181--185, 1987.</ref>

</references>