Reduction (mathematics) - Static (Guyan) Reduction

Static (Guyan) Reduction

In dynamic analysis, Static Reduction refers to reducing the number of degrees of freedom. Static Reduction can also be used in FEA analysis to simplify a linear algebraic problem. Since a Static Reduction requires several inversion steps it is an expensive matrix operation and is prone to some error in the solution. Consider the following system of linear equations in an FEA problem

$\begin{bmatrix} K_{11} & K_{12} \\ K_{21} & K_{22} \end{bmatrix}\begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}=\begin{bmatrix} F_{1} \\ F_{2} \end{bmatrix}$

Where K and F are known and K, x and F are divided into submatrices as shown above. If F₂ contains only zeros, and only x₁ is desired, K can be reduced to yield the following system of equations

$\begin{bmatrix} K_{11,reduced} \end{bmatrix}\begin{bmatrix} x_{1} \end{bmatrix}=\begin{bmatrix} F_{1} \end{bmatrix}$

K_11,reduced is obtained by writing out the set of equations as follows

$K_{11}x_{1}+K_{12}x_{2}=F_{1}$

$K_{21}x_{1}+K_{22}x_{2}=0$

Equation (2) can be rearranged

$-K_{22}^{-1}K_{21}x_{1}=x_{2}$

And substituting into (1)

$K_{11}x_{1}-K_{12}K_{22}^{-1}K_{21}x_{1}=F_{1}$

In matrix form

$\begin{bmatrix} K_{11}-K_{12}K_{22}^{-1}K_{21} \end{bmatrix}\begin{bmatrix} x_{1} \end{bmatrix}=\begin{bmatrix} F_{1} \end{bmatrix}$

And

$K_{11,reduced}=K_{11}-K_{12}K_{22}^{-1}K_{21}$

In a similar fashion, any row/column i of F with a zero value may be eliminated if the corresponding value of x_i is not desired. A reduced K may be reduced again. As a note, since each reduction requires an inversion, and each inversion is a n3 most large matrices are pre-processed to reduce calculation time.

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