Windscreen

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Description

Figure 1

Figure 2

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

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

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 xx--CrossReference--dft--fig1--xx. 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 $m=400$.

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:

$\begin{array}{rl}Kx-{\omega }^{2}Mx& =B\\ y& =B^{⊺}x\end{array}$

System dimensions:

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