Fast exact linear and non-linear structural reanalysis and the Sherman-Morrison-Woodbury formulas

Several exact fast static structural reanalysis techniques, introduced by researchers mostly for truss structures and some for frames and plate structures, are reviewed. Most utilize the property that the solution of a system of linear equations can be updated inexpensively when the matrix is changed by a low-rank increment. This paper shows that these methods are variants of the well-known Sherman{Morrison and Woodbury (SMW) formulas for the update of the inverse of a matrix. In addition, the paper extends the low-cost linear reanalysis in the spirit of the SMW formulas to some non-linear reanalysis problems. For a linear reanalysis, the extension reduces to the SMW formulas