Condensation Reduction

A theory of condensation model reduction establishes conditions under which degrees-of-freedom can be eliminated from semi-discrete models while retaining response-prediction fidelity in those degrees-of-freedom that remain. In fact, the full-degree-of-freedom prediction for the original model can be recovered from the corresponding prediction of the reduced-degree-of-freedom model. Since continuum models can be made semi-discrete by common, well-understood techniques, the method has broad applicability. By design, the method is directly implementable on a computer and easily interfaces with current computational methods such as finite elements. A general condensation scheme is given first and then specialized to the condensation of generic linear and quadratically nonlinear dynamic models, the extension to higher order polynomial nonlinearities being straightforward. Exact results are obtained for the constant coefficient linear dynamic case. As an application, degree-of-freedom reduction in a spatially discretized model of a deterministic, heterogeneous material can be made to correspond to homogenization/smoothing of that material's behavior. In contrast to the multiple scales and similar homogenization/smoothing methods, the condensation method does not make use of a periodic media assumption and it fully and directly incorporates boundary conditions. In continuous-frequency, spatially-discrete applications, such as the structural acoustics of large, complex systems with realistic, finite-element-modeled geometries, the condensation method can target specific regions of the spectrum, not necessarily near zero frequency, for which one would like frequency-response fidelity. It can also function as an alternative to finite 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.