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I have a large structure with many DOFs. But the application of force is very localized say over one or two nodes. If I try to run FEA on this with Guyan Reduction, I think the efficiency of reduction would be very less.

The stiffness matrix can be partitioned into master and slave stiffness matrices:
$K = \begin{bmatrix} K_{mm} & K_{ms} \\ K_{sm} & K_{ss} \end{bmatrix}$

Now in my scenario, $K_{ss}$ will be very large and we have to invert it anyhow to get the reduced stiffness matrix:
$K_{reduced} = K_{mm} - K_{ms} K_{ss}^{-1} K_{sm}$

Is there any condensation techniques to avoid this and how to implement it for this kind of problem?

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  • $\begingroup$ A technique to avoid what? Inverting $K_{ss}$? $\endgroup$ Jan 25 at 15:21
  • $\begingroup$ @WolfgangBangerth: Yes, because size of $K_{ss}$ is as big as $K$ itself. $\endgroup$
    – s6292_1997
    Jan 25 at 18:24
  • $\begingroup$ In that case, just don't invert $K_{ss}$ and solve with all of $K$ instead. $\endgroup$ Jan 26 at 0:16

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If $m$ is a small number compared to $s$, you do not need to invert $K_{ss}$ to calculate the Schur complement.

You just need to calculate all $m$ columns of the $s\times m$ matrix $K_{ss}^{-1}K_{sm}$ by solving $m$ linear systems with $K_{ss}$ as the system matrix, and each column of the $K_{sm}$ matrix as the right-hand-side.

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