(Init Windscreen) |
(Added dimension section and fixes) |
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==Description== |
==Description== |
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<figure id="fig1">[[File:Windscreen1.gif|490px|thumb|right|Figure 1]]</figure> |
<figure id="fig1">[[File:Windscreen1.gif|490px|thumb|right|Figure 1]]</figure> |
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| − | <figure id=" |
+ | <figure id="fig2">[[File:Windscreen2.png|490px|thumb|right|Figure 2]]</figure> |
This is an example for a model in the frequency domain of the form |
This is an example for a model in the frequency domain of the form |
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The test problem is a structural model of a car windscreen. |
The test problem is a structural model of a car windscreen. |
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This is a 3D problem discretized with <math>7564</math> nodes and <math>5400</math> linear hexahedral elements (3 layers of <math>60 \times 30</math> elements). |
This is a 3D problem discretized with <math>7564</math> nodes and <math>5400</math> linear hexahedral elements (3 layers of <math>60 \times 30</math> elements). |
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| − | The mesh is shown in |
+ | The mesh is shown in <xr id="fig1"/>. |
The material is glass with the following properties: |
The material is glass with the following properties: |
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The Young modulus is <math>7\times10^{10}\mathrm{N}/\mathrm{m}^2</math>, the density is <math>2490 \mathrm{kg}/\mathrm{m}^3</math>, and the Poisson ratio is <math>0.23</math>. The natural damping is <math>10\%</math>, i.e. <math>\gamma=0.1</math>. |
The Young modulus is <math>7\times10^{10}\mathrm{N}/\mathrm{m}^2</math>, the density is <math>2490 \mathrm{kg}/\mathrm{m}^3</math>, and the Poisson ratio is <math>0.23</math>. The natural damping is <math>10\%</math>, i.e. <math>\gamma=0.1</math>. |
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The structural boundaries are free (free-free boundary conditions). |
The structural boundaries are free (free-free boundary conditions). |
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The windscreen is subjected to a point force applied on a corner. |
The windscreen is subjected to a point force applied on a corner. |
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| − | The goal of the model reduction is the fast evaluation of y. |
+ | The goal of the model reduction is the fast evaluation of <math>y</math>. |
Model reduction is used as a fast linear solver for a sequence of parametrized linear systems. |
Model reduction is used as a fast linear solver for a sequence of parametrized linear systems. |
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\{0.5j,j=1,\ldots,m\}</math> with <math>m=400</math>. |
\{0.5j,j=1,\ldots,m\}</math> with <math>m=400</math>. |
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| − | + | <xr id="fig1"/> and <xr id="fig2"/> show the mesh of the car windscreen and frequency response function. |
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==Origin== |
==Origin== |
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| − | This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>. |
+ | This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38886. |
==Data== |
==Data== |
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The archive contains files <tt>windscreen.K</tt>, <tt>windscreen.M</tt> and <tt>windscreen.B</tt> representing <math>Kd</math>, <math>M</math> and <math>f</math> accordingly. |
The archive contains files <tt>windscreen.K</tt>, <tt>windscreen.M</tt> and <tt>windscreen.B</tt> representing <math>Kd</math>, <math>M</math> and <math>f</math> accordingly. |
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| + | |||
| + | ==Dimensions== |
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| + | |||
| + | System structure: |
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| + | |||
| + | :<math> |
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| + | \begin{align} |
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| + | K x - \omega^2 M x &= B \\ |
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| + | y &= B^\intercal x |
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| + | \end{align} |
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| + | </math> |
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| + | |||
| + | System dimensions: |
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| + | |||
| + | <math>K \in \mathbb{R}^{22692 \times 22692}</math>, |
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| + | <math>M \in \mathbb{R}^{22692 \times 22692}</math>, |
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| + | <math>B \in \mathbb{R}^{22692 \times 1}</math>. |
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==References== |
==References== |
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Revision as of 12:38, 1 March 2018
Note: This page has not been verified by our editors.
Description
This is an example for a model in the frequency domain of the form
\( \begin{array}{rcl} K_d x - \omega^2 M x & = & f \\ y & = & f^* x \end{array} \)
where \(f\) represents a unit point load in one unknown of the state vector. \(M\) is a symmetric positive-definite matrix and \(K_d = (1+i\gamma) K\) where \(K\) is symmetric positive semi-definite.
The test problem is a structural model of a car windscreen. This is a 3D problem discretized with \(7564\) nodes and \(5400\) linear hexahedral elements (3 layers of \(60 \times 30\) elements). The mesh is shown in xx--CrossReference--dft--fig1--xx. The material is glass with the following properties: The Young modulus is \(7\times10^{10}\mathrm{N}/\mathrm{m}^2\), the density is \(2490 \mathrm{kg}/\mathrm{m}^3\), and the Poisson ratio is \(0.23\). The natural damping is \(10\%\), i.e. \(\gamma=0.1\). The structural boundaries are free (free-free boundary conditions). The windscreen is subjected to a point force applied on a corner. The goal of the model reduction is the fast evaluation of \(y\). Model reduction is used as a fast linear solver for a sequence of parametrized linear systems.
The discretized problem has dimension \(n=22692\). The goal is to estimate \(x(\omega)\) for \(\omega\in[0.5,200]\). In order to generate the plots the frequency range was discretized as \(\{\omega_1,\ldots,\omega_m\} = \{0.5j,j=1,\ldots,m\}\) with \(m=400\).
xx--CrossReference--dft--fig1--xx and xx--CrossReference--dft--fig2--xx show the mesh of the car windscreen and frequency response function.
Origin
This benchmark is part of the Oberwolfach Benchmark Collection[1]; No. 38886.
Data
Download matrices in the Matrix Market format:
- windscreen.tar.gz (21.5 MB)
The archive contains files windscreen.K, windscreen.M and windscreen.B representing \(Kd\), \(M\) and \(f\) accordingly.
Dimensions
System structure:
\[ \begin{align} K x - \omega^2 M x &= B \\ y &= B^\intercal x \end{align} \]
System dimensions\[K \in \mathbb{R}^{22692 \times 22692}\], \(M \in \mathbb{R}^{22692 \times 22692}\), \(B \in \mathbb{R}^{22692 \times 1}\).
References
- ↑ J.G. Korvink, E.B. Rudnyi, Oberwolfach Benchmark Collection, Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 311--315, 2005.