A THEORY OF CONDENSATION MODEL-REDUCTION

A theory of condensation model reduction establishes conditions under which degrees-of-freedom can be eliminated from semi-discrete models w hile retaining response-prediction fidelity in those degrees-of-freedo m that remain. In fact, the full-degree-of-freedom prediction for the original model can be recovered from the corresponding prediction of t he reduced-degree-of-freedom model. Since continuum models can be made semi-discrete by common, well-understood techniques, the method has b road applicability. By design, the method is directly implementable on a computer and easily interfaces with current computational methods s uch as finite elements. A general condensation scheme is given first a nd then specialized to the condensation of generic linear and quadrati cally nonlinear dynamic models, the extension to higher order polynomi al nonlinearities being straightforward. Exact results are obtained fo r the constant coefficient linear dynamic case. As an application, deg ree-of-freedom reduction in a spatially discretized model of a determi nistic, heterogeneous material can be made to correspond to homogeniza tion/smoothing of that material's behavior. In contrast to the multipl e scales and similar homogenization/smoothing methods, the condensatio n method does not make use of a periodic media assumption and it fully and directly incorporates boundary conditions. In continuous-frequenc y, spatially-discrete applications, such as the structural acoustics o f large, complex systems with realistic, finite-element-modeled geomet ries, the condensation method can target specific regions of the spect rum, not necessarily near zero frequency, for which one would like fre quency-response fidelity. It can also function as an alternative to fi nite element modal decomposition without the accompanying restrictions on damping. For eigenvalue problems it is shown that all eigenvalues of the reduced model are also those of the original model. In addition , an eigenvalue economizer condensation method in current use is shown to be an almost trivial special case of this approach.