Bond strength uncertainty quantification via confidence intervals for nondestructive evaluation of bonded composites

As bonded composite materials are used more frequently for aerospace applications, it is necessary to certify that parts achieve desired levels of certain physical characteristics (e.g., strength) for safety and performance. Nondestructive evaluation (NDE) of adhesively bonded structures enables verification of bond physical characteristics, but uncertainty quantification (UQ) of NDE estimates is crucial for understanding risks, especially for NDE estimates like bond strength. To address the critical need for NDE UQ for adhesive bond strength estimates, we propose an optimization–based approach to computing finite–sample confidence intervals showing the range of bond strengths that could feasibly be produced by the observed data. A statistical inverse model approach is used to compute a confidence interval of specimen interfacial stiffness from swept–frequency ultrasonic phase observations and a method for propagating the interval to bond strength via a known interfacial stiffness regression is proposed. This approach requires innovating the optimization–based confidence interval to handle both a nonlinear forward model and unknown variance and developing a calibration approach to ensure that the final bond strength interval achieves at least the desired coverage level. Using model assumptions in line with current literature, we demonstrate our approach on simulated measurement data using a variety of low to high noise settings under two prototypical parameter settings. Relative to a baseline approach, we show that our method achieves better coverage and smaller intervals in high–noise settings and when a nuisance parameter is near the constraint boundary.

💡 Research Summary

This paper addresses a critical challenge in the certification of adhesively bonded composite structures for aerospace applications: the nondestructive evaluation (NDE) and uncertainty quantification (UQ) of bond strength. While ultrasonic techniques can provide measurements related to bond quality, translating these signals into a quantitative, probabilistic estimate of mechanical strength with guaranteed reliability has been a significant hurdle.

The proposed methodology builds upon prior work that established a linear statistical relationship between interfacial stiffness (derivable from ultrasonic measurements) and ultimate bond strength (obtained via mechanical testing). The core innovation lies in adding a rigorous UQ layer to this pipeline. The approach is a two-stage, frequentist, optimization-based framework.

In the first stage, a confidence interval for the interfacial stiffness of a specific specimen is constructed from swept-frequency ultrasonic phase data. This is achieved using an optimization-based confidence interval method. Instead of yielding a single point estimate, this method defines a “confidence set” in the full parameter space (including interfacial stiffness, attenuation, bond thickness, etc.) containing all parameter vectors consistent with the observed data within a statistically calibrated tolerance. The interval for the quantity of interest (interfacial stiffness) is then computed as the minimum and maximum values of that parameter within this set. The authors extend existing theory to handle the nonlinear forward physics model (relating parameters to phase measurements) and the practically relevant case of unknown measurement error variance.

In the second stage, this interfacial stiffness confidence interval is propagated through the pre-established linear regression model (relating interfacial stiffness to bond strength) to produce a final confidence interval for bond strength. A calibration procedure is developed to ensure this final interval maintains the desired coverage probability (e.g., 95%) despite the propagation step.

The method’s performance is validated through comprehensive numerical simulations under two prototypical parameter settings (“typical” and “boundary,” where a nuisance parameter is near its physical constraint) and across a spectrum of low to high noise levels. It is compared against a baseline Bayesian approach. The results demonstrate that the proposed method achieves superior performance, particularly in challenging high-noise scenarios and for the “boundary” parameter case. It provides confidence intervals that are both more reliable (achieving coverage closer to the nominal 95% level) and more precise (shorter average interval length) than the baseline. This indicates that the method successfully avoids being overly conservative while still providing strong statistical guarantees.

A key philosophical advantage highlighted is the frequentist interpretation: when deployed repeatedly to certify many parts, the procedure guarantees that the true bond strength will fall within the calculated interval at least (1-α)*100% of the time. Furthermore, the method natively incorporates known physical constraints on parameters (e.g., positive stiffness, bounded thickness), leveraging this information to tighten intervals without biasing their statistical validity. The work concludes that this optimization-based confidence interval approach provides a robust, interpretable, and practical foundation for quantitative NDE and risk-informed decision-making in the certification of bonded composite structures.