Current dynamic substructuring methods as approximations to condensation model reduction

Abstract

Condensation Model Reduction (CMR) theory, when viewed as a dynamic substructuring method, is shown to encompass much of the existing dynamic substructuring methods as special cases of a single unified approach. Dynamic substructuring refers to the partitioning of a semi-discrete (continuous-in-time) mathematical model with respect to its dependence on the discretized independent variables (usually spatial) with the typical intention of eliminating most of the degrees-of-freedom in each partition (substructure) of a subset of isolated partitions. As an example, a contiguous subset of finite elements could be viewed as a substructure (superelement). One important use of dynamic substructuring is, hence, to analyze complex dynamic systems that are too large for current computers by reducing their size. The three currently used methods of dynamic substructuring are referred to as Guyan, Improved Reduced System (IRS), and Component Mode Synthesis (CMS) reduction, the latter having several variants. (The related Modal Reduction method was not considered here since, in that method, the work associated with reducing a particular substructure is not limited to that substructure.) In contrast to the current methods, the accuracy of CMR can be systematically improved by the inclusion of nonmodal higher order terms (as well as by the inclusion of additional modes). It is shown that the transformation matrices associated with both IRS and the Craig-Bampton (fixed interface) version of CMS approximate corresponding special cases of CMR. (The reduced-degree-of-freedom mass, damping, and stiffness matrices are different from those of CMR because CMR does not reduce the matrices via a matrix transformation.) In addition, the appearance of modes in CMR occurs as a natural consequence of the theory and not, as in CMS, as a heuristic inclusion to the transformation matrix. Classic Guyan reduction, in its entirety, is also shown to approximate a special case of CMR.