Direct Reanalysis

Direct (closed form) reanalysis methods are efficient for low-rank changes in the stiffness matrix. In particular, these methods are applicable to situations where a relatively small proportion of the structure is changed and the changes in the stiffness matrix can be represented by a small submatrix. These methods are inefficient when the sub-matrix of changes in the system stiffness matrix is of high-rank or large.

Direct methods are usually based on the Sherman-Morrison [1] and Woodbury [2] formulae for the update of the inverse of a matrix. Surveys on these methods are given elsewhere [3–5]. A comprehensive historical survey of the origin of these formulae is presented in [4]. It has been shown [5] that various reanalysis methods may be viewed as variants of these formulae. When the stiffness matrix is modified by a rank-one increment, the solution can be updated inexpensively with the Sherman-Morrison formula by solving the initial analysis equations with a different right-hand side vector, which is a factor of the matrix increment. Similarly, solution for a higher-rank change in the stiffness matrix can be carried out by superposition of rank-one changes. This is reflected in the Woodbury formula.

Direct reanalysis methods are presented in Sect. 8.1. The Sherman- Morrison formula for exact solutions for a single rank-one change is introduced in Sect. 8.1.1, the Woodbury formula for multiple rank-one changes is discussed in Sect. 8.1.2, and a procedure for general changes in the design is presented in Sect. 8.1.3. In Sect. 8.2 direct solutions, which are based on the CA approach, are developed. For multiple rank-one changes in the stiffness matrix the presented CA solution procedure and the Sherman-Morrison and Woodbury formulae are equivalent. It is shown in Sect. 8.3 that direct solutions can be obtained by the CA approach also for topological and geometrical changes.