Example - Local pMOR by Interpolation of Reduced System Matrices

Problem

The given cantilevered beam has a quadratic cross section ($//<![CDATA[ \begin{array}{l}a=\SI{0.01}{\meter}\end{array} //]]>$) with $//<![CDATA[ \begin{array}{l}E=\SI{2.1e11}{\newton\per\meter\squared}, \nu=0.3, \rho=\SI{7860}{\kilogram\per\meter\cubed}\end{array} //]]>$. It is discretized using the Finite Element Method and excited at the tip with $//<![CDATA[ \begin{array}{l}F=\SI{1}{\newton}\end{array} //]]>$ and of varying circular frequency ($//<![CDATA[ \begin{array}{l}\Omega=\frac{s}{\mathrm{i}}\end{array} //]]>$). No damping is assumed.

Generating a Reduced Order Model (ROM) via Modal Truncation with $//<![CDATA[ \begin{array}{l}r=15\end{array} //]]>$ for a length of $//<![CDATA[ \begin{array}{l}l=\SI{1.1}{\meter}\end{array} //]]>$ leads to a good approximation

Now, we want to generate a parametric Reduced Order Model. Therefore, we compute the reduced matrices for two samples: $//<![CDATA[ \begin{array}{l}\mathbf{p}_1(l=\SI{1.0}{\meter})\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\mathbf{p}_2(l=\SI{1.2}{\meter})\end{array} //]]>$. To motivate the need for a transformation, we directly interpolate the reduced matrices obtained for the two samples to get reduced matrices for the queried parameter point $//<![CDATA[ \begin{array}{l}\mathbf{p}(l=\SI{1.1}{\meter})\end{array} //]]>$. 

It can be seen that this gives very inaccurate results, especially for higher frequencies.

Before the transformation, most modes have the same shape but are deflected in opposite directions or change their position.

To circumvent this problem we now first compute a generalized coordinate system $//<![CDATA[ \begin{array}{l}\mathbf{R}\end{array} //]]>$ and transform the reduced matrices from the individual samples to this coordinate system following the approach of pMOR by Matrix Interpolation.

After the transformation the modes resemble the generalized coordinate system and each other.

This delivers a very accurate approximation of the full solution also for higher frequencies. Note that errors still occur because the reduced system matrices for the queried parameter point are obtained by linear interpolation but the dependency of the stiffness matrix on the length of the beam is cubic.