A Stochastic Projection-Based Hyperreduced Order Model for Model-Form Uncertainties in Vibration Analysis

Abstract

A feasible, nonparametric, probabilistic approach for quantifying model-form uncertainties associated with a High-Dimensional computational Model (HDM) and/or a corresponding Hyperreduced Projection-based Reduced-Order Model (HPROM) designed for the solution of generalized eigenvalue problems arising in vibration analysis, is presented. It is based on the construction of a Stochastic HPROM (SHPROM) associated with the HDM and its HPROM using three innovative ideas: the substitution of the deterministic Reduced-Order Basis (ROB) with a Stochastic counterpart (SROB) that features a reduced number of hyperparameters; the construction of this SROB on a compact Stiefel manifold in order to guarantee the linear independence of its column vectors and the satisfaction of any applicable constraints; and the formulation and solution of a reduced-order inverse statistical problem to determine the hyperparameters so that the mean value and statistical fluctuations of the eigenvalues predicted in real time using the SHPROM match target values obtained from available data. If the data are experimental data, the proposed approach models and quantifies the model-form uncertainties associated with the HDM, while accounting for the modeling errors introduced by model reduction. If on the other hand the data are high-dimensional numerical data, the proposed approach models and quantifies the model-form uncertainties associated with the HPROM. Consequently, the proposed nonparametric, probabilistic approach for modeling and quantifying model-form uncertainties can also be interpreted as an effective means for extracting fundamental information or knowledge from data that is not captured by a deter-ministic computational model, and incorporating it in this model. Its potential for quantifying model-form uncertainties in eigenvalue computations is demonstrated for what-if? vibration analysis scenarios associated with shape changes for a jet engine nozzle.