Windscreen

Description

Figure 1

Figure 2

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

\[ \begin{align} K_d x - \omega^2 M x & = f \\ y & = f^* x \end{align} \]

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 shows the mesh of the car windscreen and xx--CrossReference--dft--fig2--xx the frequency response \(\vert \Re(y(\omega)) \vert\).

Origin

This benchmark is part of the Oberwolfach Benchmark Collection; 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 \(K_d\), \(-M\) and \(f\) accordingly.

Dimensions

System structure: \[ \begin{align} (K + \omega^2 M) x & = B \\ y & = B^{\mathrm{T}} x \end{align} \] 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, 2004. 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 =         2004
}

• For the background on the benchmark:

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