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:
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
is obtained by writing out the set of equations as follows:
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(1) |
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(2) |
Equation (2) can be solved for (assuming invertibility of ):
And substituting into (1) gives
Thus
In a similar fashion, any row or column i of F with a zero value may be eliminated if the corresponding value of xi 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.