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 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; 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\[K \in \mathbb{R}^{22692 \times 22692}\], \(M \in \mathbb{R}^{22692 \times 22692}\), \(B \in \mathbb{R}^{22692 \times 1}\).