Windscreen

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Description

Figure 1

Figure 2

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 Figure 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 \(m=400\).

Figure 1 and Figure 2 show the mesh of the car windscreen and frequency response function.

Origin

This benchmark is part of the Oberwolfach Benchmark Collection.

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.