Parameterized Reduced Order Models Constructed Using Hyper Dual Numbers

Abstract

In order to assess the predicted performance of a manufactured system, analysts must typically consider random variations (both geometric and material) in the development of a finite element model, instead of a single deterministic model of an idealized geometry. The incorporation of random variations, however, could potentially require the development of thousands of nearly identical solid geometries that must be meshed and separately analyzed, which would require an impractical number of man-hours to complete. This paper proposes a new approach to uncertainty quantification by developing parameterized reduced order models. These parameterizations are based upon Taylor series expansions of the system’s matrices about the ideal geometry, and a component mode synthesis representation for each linear substructure is used to form an efficient basis with which to study the system. The numerical derivatives required for the Taylor series expansions are obtained efficiently using hyper dual numbers, which enable the derivatives to be calculated precisely to within machine precision. The theory is applied to a stepped beam system in order to demonstrate proof of concept. The accuracy and efficiency of the method, as well as the level at which the parameterization is introduced, are discussed. Hyper dual numbers can be used to construct parameterized models both efficiently and accurately and constitute an appropriate methodology to account for perturbations in a structural system.