Parameterized reduced order models from a single mesh using hyper-dual numbers

In order to assess the predicted performance of a manufactured system, analysts must consider random variations (both geometric and material) in the development of a model, instead of a single deterministic model of an idealized geometry with idealized material properties. The incorporation of random geometric variations, however, potentially could necessitate 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 research advances a recent 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 via hyper-dual numbers, and are compared to parameterized models constructed with finite difference formulations. The advantage of using hyper-dual numbers is two-fold: accuracy of the derivatives to machine precision, and the need to only generate a single mesh of the system of interest. The theory is applied to a stepped beam system in order to demonstrate proof of concept. The results demonstrate that the hyper-dual number multivariate parameterization of geometric variations, which largely are neglected in the literature, are accurate for both sensitivity and optimization studies. As model and mesh generation can constitute the greatest expense of time in analyzing a system, the foundation to create a parameterized reduced order model based off of a single mesh is expected to reduce dramatically the necessary time to analyze multiple realizations of a component׳s possible geometry.