Windscreen

Windscreen

Background
Benchmark ID	windscreen_n22692m1q1
Category	oberwolfach
System-Class	LTI-SOS
Parameters
nstates	22692
ninputs	1
noutputs	1
nparameters	0
components	B, C, K, M
Copyright
License	NA
Creator	Christian Himpe
Editor	Christian Himpe Petar Mlinarić Yao Yue
Location
NA

Description

Figure 1

Figure 2

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

\[ \begin{align} K x + \omega^2 M x & = B \\ y & = C x \end{align} \]

where \(B\) represents a unit point load in one unknown of the state vector, \(C = B^T \), \(M\) is a symmetric positive-definite matrix, and \(K = (1+i\gamma) \widetilde{K}\) with \(\widetilde{K}\) symmetric positive semidefinite.

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 \(7\times10^{10}\mathrm{N}/\mathrm{m}^2\), 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 \(m=400\).

Fig. 1 shows the mesh of the car windscreen and Fig. 2 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-dim1e4-windscreen.tar.gz (21.5 MB)

The archive contains files windscreen.K, windscreen.M and windscreen.B representing \(K\), \(M\) and \(B\) accordingly.

Dimensions

System structure: \[ \begin{align} (K + \omega^2 M) x & = B \\ y & = C x \end{align} \] with \(\omega \in [0.5, 200]\).

System dimensions\[K \in \mathbb{C}^{22692 \times 22692}\], \(M \in \mathbb{R}^{22692 \times 22692}\), \(B \in \mathbb{R}^{22692 \times 1}\), \(C \in \mathbb{R}^{1 \times 22692}\),

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. https://modelreduction.org/morwiki/Windscreen

@MISC{morwiki_windscreen,
  author =       {{Oberwolfach Benchmark Collection}},
  title =        {Windscreen},
  howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
  url =          {https://modelreduction.org/morwiki/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},
}