Improved model correlation through optimal parameter ranking using model reduction algorithms: Augmenting engineering judgment

Abstract

As the complexity and scales of dynamic models increase, novel and efficient model correlation methodologies are vital to the development of accurate models. Classically, to correlate a Finite Element Model (FEM) such that it matches a dynamic test, an experienced engineer chooses a small subset of input parameters that are surmised to be crucial, sensitive and/or possibly erroneous. The operator will then use engineering judgment, or a model updating technique to update the selected subset of parameters until the error between the FEM and the test article is reduced to within a set bound. To reduce the intricacy and difficulty of model correlation, a methodology is proposed to provide a quantitative parameter importance ranking using a model reduction algorithm applied to a parameter sensitivity analysis. Four model reduction algorithms are studied in this effort, the Discrete Empirical Interpolation Method (SVD-DEIM), Q-DEIM, Projection Coefficient and finally Weighted Projection Coefficient. These model reduction methods identify and rank critical parameters, enabling the selection of a minimum set of critical correlation parameters. This reduced set of parameters results in reduced computational resources and engineering effort required to generate a correlated model. The insight gained using these methods is essential in developing an optimal, reduced parameter set that provides high correlation capability with minimal iterative costs. To evaluate the proposed parameter selection methodology, a representative set of academic and industry experts provided their engineering judgment for comparison with the methodology presented. A comprehensive investigation of the robustness of this methodology is performed on a simple cantilever beam for demonstration. The scale of the model has expressly been chosen to allow for all potential ranking variations to be evaluated so that these ranking methods can be understood relative to the true optimal ranking. The ranking robustness to incorrect engineering judgment, resulting in uncertainty in the assumed size of the design space and, therefore, the error bounds, is investigated. The methodology presented identifies the most useful parameters for correlation, enabling a straightforward and computationally efficient model correlation approach as compared with other methods. To quantify the ranking quality, a metric, the Correlation Norm Error, is developed. For the problem discussed, blind random assessments result in a Correlation Norm Error of 413.3. Engineering judgment has been shown to improve upon blind random assessments, reducing the Correlation Norm Error to 334.3. The best performing model reduction method, Q-DEIM using 10 FEM runs as the input, was able to identify the optimal ranking correctly, reducing the Correlation Norm Error to zero.

The discrete empirical interpolation method in class identification and data summarization

“…While still in the modeling context, Phoenix et al (2018) use DEIM and Q-DEIM to rank parameters for the purposes of performing model correlation in validating differential equations models. The authors perform a wide array of experiments, forming Sensitivity matrices resulting from discretized differential equations models with different parameter inputs.…”

“…As with many dimension reduction techniques, the choice for $$$ k $$$ (and/or $$$ \hat{k} $$$ in the oversampling problem) is not always clear a priori. For instance, Sorensen and Embree (2016) find leverage score rankings move around as $$$ k $$$ grows, and Phoenix et al (2018) present results showing a sweet spot for DEIM/Q‐DEIM applied to more moderate Sensitivity matrix sizes rather than larger matrices with additional data—more information does not necessarily imply better outcomes. It is also worth noting that even though DEIM has been shown to perform well in reducing the dataset while capturing members of most classes (both prevalent and rare‐occurring), some reduced sets still contain several data points with the same class label.…”

“…In comparison to other evaluated model reduction methods (and a panel of professional engineers), Q‐DEIM and DEIM both out‐perform the other techniques with statistical significance, including in cases with larger uncertainties in the parameter value ranges. In fact, although there is not statistical significance between the reported DEIM and Q‐DEIM results, the implementation of Q‐DEIM is able to identify the exhaustively identified optimal parameter ranking for the test problem (Phoenix et al, 2018). The Q‐DEIM parameter selection ideas presented in this work for sensitivity analysis are then applied in studying morphing waverider design in identifying important actuator locations on the morphing surface (Phoenix et al, 2019).…”