# FEA with order reduced for a large system with a very localized force application

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?

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

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.