Windscreen

Description

Figure 1

Figure 2

This is an example for a model in the frequency domain of the form

$\begin{array}{rl}{K}_{d}x-{\omega }^{2}Mx& =f\\ y& =f^{*}x\end{array}$

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 Fig. 1. The material is glass with the following properties: The Young modulus is $7\times 10^{10}\mathrm{N}/{\mathrm{m}}^2$, the density is $2490\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$, and the Poisson ratio is $0.23$. The natural damping is $10\mathrm{\%}$, 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 $\{{\omega }_1,\dots ,{\omega }_m\}=\{0.5j,j=1,\dots ,m\}$ with $m=400$.

Fig. 1 shows the mesh of the car windscreen and Fig. 2 the frequency response $|\mathrm{\Re }(y(\omega ))|$.

Origin

This benchmark is part of the Oberwolfach Benchmark Collection; No. 38886.

Data

Download matrices in the Matrix Market format:

• Windscreen-dim1e4-windscreen.tar.gz (21.5 MB)

The archive contains files windscreen.K, windscreen.M and windscreen.B representing $K_d$, $-M$ and $f$ accordingly.

Dimensions

System structure:

$\begin{array}{rl}(K+{\omega }^{2}M)x& =B\\ y& =B^{\mathrm{T}}x\end{array}$

with $\omega \in [0.5,200]$.

System dimensions:

$K\in {\mathbb{ℂ}}^{22692\times 22692}$, $M\in {\mathbb{ℝ}}^{22692\times 22692}$, $B\in {\mathbb{ℝ}}^{22692\times 1}$.

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

Oberwolfach Benchmark Collection, Windscreen. hosted at MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Windscreen

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

• For the background on the benchmark:

@article{Mee07,
  author =       {K. Meerbergen},
  title =        {Fast frequency response computation for {R}ayleigh damping},
  journal =      {International Journal for Numerical Methods in Engineering},
  volume =       {73},
  number =       {1},
  pages =        {96--106},
  year =         {2007},
  doi =          {10.1002/nme.2058},
}