Condensation

Static and Dynamic Condensation are two methods used for Model Order Reduction. They are used for complex and large numerical models, which require a substantial computational effort and large amounts of memory. Model order reduction methods reduce the size of the numerical models. The reduced models approximate the full system's response characteristics with less computational effort and memory requirements. Good reduced models are able to replace the original model.

For Static and Dynamic Condensation, the reduction is done using physical coordinates. This means that a suitable subset of physical coordinates is removed from the model.

Static Condensation

Consider the static equilibrium

and partition the system into a set of active dofs $//<![CDATA[ \begin{array}{l}\mathbf{x}_a\in\mathbb{R}^n\end{array} //]]>$ which will be retained and inactive dofs $//<![CDATA[ \begin{array}{l}\mathbf{x}_i\in\mathbb{R}^m\end{array} //]]>$ which will be condensed:

By splitting the equilibrium equation, $//<![CDATA[ \begin{array}{l}\mathbf{x}_i\end{array} //]]>$ can be expressed depending on $//<![CDATA[ \begin{array}{l}\mathbf{x}_a\end{array} //]]>$:

The substitution of $//<![CDATA[ \begin{array}{l}\mathbf{x}_i\end{array} //]]>$ into the first line of the static equilibrium yields a system of equations of size $//<![CDATA[ \begin{array}{l}n\end{array} //]]>$

For $//<![CDATA[ \begin{array}{l}\mathbf{f}_i=0\end{array} //]]>$, this substitution can also be expressed using matrix notation

By premultiplying with $//<![CDATA[ \begin{array}{l}\mathbf{V}^H = \begin{bmatrix} \mathbf{I} & -\mathbf{K}_{ii}^{-1}\mathbf{K}_{ia} \end{bmatrix}\end{array} //]]>$ we get the transformation from full to active coordinates

This is the matrix notation of the previous equation (1)

and can be written as a reduced system as

Under this premises, static condensation can be considered as a model order reduction method regarding the reduction basis $//<![CDATA[ \begin{array}{l}\mathbf{V}\end{array} //]]>$. Note, that the reduced model yields the exact same results as the full model.

An Exercise for Static Condensation can be found here:

• Try it! - Static Condensation

Dynamic Condensation

Compared to Static Condensation, now also inertia effects are considered:

Using the same approach as in the static condensation, the system is partitioned in a set of active and inactive dofs and reduced with

Due to the inertia effects, the reduced system does not yield the exact same results as the full system. The accuracy of the reduction is highly dependent on the physical behavior of the system: one should choose dofs with significant mass and/or displacement as active degrees of freedom.

Further reading: Chap.~7 in (Qu, 2004), (Friswell et al, 1995)

An example for Dynamic Condensation can be found here:

• Example - Dynamic Condensation

References

Friswell, M. I., Garvey, S. D., & Penny, J. E. T. (1995). Model reduction using dynamic and iterated IRS techniques. Journal of sound and vibration, 186(2), 311-323.

Qu, Z. Q. (2004). Model order reduction techniques with applications in finite element analysis: with applications in finite element analysis. Springer Science & Business Media.