Reduction (mathematics)

In dynamic analysis, static reduction refers to reducing the number of degrees of freedom. Static reduction can also be used in finite element analysis to refer to simplification of 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:

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

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

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

${\displaystyle K_{11,{\text{reduced}}}}$ is obtained by writing out the set of equations as follows:

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

1

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

2

Equation (2) can be solved for ${\displaystyle x_{2}}$ (assuming invertibility of ${\displaystyle K_{22}}$):

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

And substituting into (1) gives

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

Thus

${\displaystyle K_{11,{\text{reduced}}}=K_{11}-K_{12}K_{22}^{-1}K_{21}.}$

In a similar fashion, any row or 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 an operation with computational cost O(n³), most large matrices are pre-processed to reduce calculation time.

In the 9th century, Persian mathematician Al-Khwarizmi's Al-Jabr introduced the fundamental concepts of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation and the cancellation of like terms on opposite sides of the equation. This is the operation which Al-Khwarizmi originally described as al-jabr.^[1] The name "algebra" comes from the "al-jabr" in the title of his book.