In computational mechanics, Guyan reduction,[1] also known as static condensation, is a dimensionality reduction method which reduces the number of degrees of freedom by ignoring the inertial terms of the equilibrium equations and expressing the unloaded degrees of freedom in terms of the loaded degrees of freedom.
The static equilibrium equation can be expressed as:
Kd=f
K
f
d
\begin{bmatrix} Kmm&Kms\\ Ksm&Kss\end{bmatrix} \begin{Bmatrix} dm\\ ds\end{Bmatrix} = \begin{Bmatrix} fm\\ fs\end{Bmatrix}
Ksmdm+Kssds=fs
Solving the above equation in terms of the independent (master) degrees of freedom leads to the following dependency relations
ds=
-1 | |
K | |
ss |
fs-
-1 | |
K | |
ss |
Ksmdm
Substituting the dependency relations on the upper partition of the static equilibrium problem condenses away the slave degrees of freedom, leading to the following reduced system of linear equations.
\left[Kmm-Kms
-1 | |
K | |
ss |
Ksm\right]dm=fm-Kms
-1 | |
K | |
ss |
fs
This can be rewritten as:
Kreduceddm=freduced
The above system of linear equations is equivalent to the original problem, but expressed in terms of the master's degrees of freedom alone.Thus, the Guyan reduction method results in a reduced system by condensing away the slave degrees of freedom.
The Guyan reduction can also be expressed as a change of basis which produces a low-dimensional representation of the original space, represented by the master's degrees of freedom.The linear transformation that maps the reduced space onto the full space is expressed as:
\begin{Bmatrix} dm\\ ds \end{Bmatrix}= \begin{bmatrix} I\\ -
-1 | |
K | |
ss |
Ksm\end{bmatrix} \begin{Bmatrix} dm \end{Bmatrix} = \begin{Bmatrix} TG \end{Bmatrix} \begin{Bmatrix} dm \end{Bmatrix}
TG
KGdm=fm
In the above equation,
KG
KG=
T | |
T | |
G |
KTG
The Guyan reduction is an integral part of the classic dynamic substructuring method known as the Craig-Bampton (CB) method. The static portion of the reduced system matrices derived from the CB method is a direct result of the Guyan reduction. It is calculated in the same manner as the Guyan stiffness matrix
KG
TG
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