Sensitivity of probability-of-failure estimates with respect to probability of detection curve parameters

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Abstract

A methodology has been developed and demonstrated that can be used to compute the sensitivity of the probability-of-failure (POF) with respect to the parameters of inspection processes that are simulated using probability of detection (POD) curves. The formulation is such that the probabilistic sensitivities can be obtained at negligible cost using sampling methods by reusing the samples used to compute the POF. As a result, the methodology can be implemented for negligible cost in a post-processing non-intrusive manner thereby facilitating implementation with existing or commercial codes. The formulation is generic and not limited to any specific random variables, fracture mechanics formulation, or any specific POD curve as long as the POD is modeled parametrically. Sensitivity estimates for the cases of different POD curves at multiple inspections, and the same POD curves at multiple inspections have been derived. Several numerical examples are presented and show excellent agreement with finite difference estimates with significant computational savings.

Highlights

► Sensitivity of the probability-of-failure with respect to the probability-of-detection curve. ►The sensitivities are computed with negligible cost using Monte Carlo sampling. ► The change in the POF due to a change in the POD curve parameters can be easily estimated.

Introduction

Nondestructive evaluation (NDE), also known as Nondestructive Testing (NDT) or Nondestructive Inspection (NDI), plays a vital role in fracture control plans. Methods such as visual, dye penetrant, ultrasonics, radiography, and eddy current are among the common inspection techniques used to ensure structural integrity [1]. The type of inspection and the times of inspection must be carefully selected to ensure safety with reasonable cost. There are a number of industries that have a long history of application of NDE methods for structural integrity. For example, applications include nuclear [2], [3], [4], petroleum [5], aircraft structures [6], [7], [8], gas turbines [9], [10], and offshore structures [11] to name a few. Rummel et al. [12] provide a summary of a number of issues related to the application of NDE methods to systems.

The efficacy of an inspection process is characterized through the POD curve [13]. The POD defines the probability of detecting a defect as a function of the size of the defect. This concept is well known and POD curves for a particular inspection process, material, etc. are developed through statistical experiments using seeded samples of various sizes, multiple inspectors, etc [8], [14]. For example, the “hit-miss” or “a^ vs. a ” opportunities for a number of inspections and a number of operators are tabulated and analyzed statistically to determine the percentiles, e.g., 50 and 95% of the POD curve [8], [15]. The percentiles are often curve fit to parametric forms such as log-logistic, lognormal, log-odds, etc. Although to date the development of a POD curve is largely based on lab experiments, computational methods are becoming more prevalent in “Model Assisted POD”, MAPOD, methods [16].

Simulation of the inspection process is usually incorporated within a probabilistic fracture mechanics analysis also known as a risk assessment. The analysis in general considers random variables of crack size and aspect ratio, material properties, loading, inspection efficacy (represented by a POD curve) and inspection times. The output conveys an estimate of the POF of the structure as a function of time or cycles with and without inspection. The inspection process and frequency is often varied toward developing an optimized inspection schedule [17], [18].

The simulation of inspection requires the assignment of a POD curve. Typically, the selection of a POD curve is based on an established catalog of POD curves pertinent to the material and inspection method under consideration. For example, POD curves for aluminum, titanium and stainless steel materials and certain geometries are cataloged in [14]. The curve fits are the best available information based upon statistical analysis of the experimental data but there always exists statistical uncertainty in the numerical values obtained. Application of existing cataloged POD curves is convenient since no new testing is required; however, there is always some uncertainty about the applicability of the lab-developed POD curves to a field inspection where issues such as access, visual acuity, cleanliness and others arise. Also, sometimes due to cost and schedule constraints, POD curves are developed for a new inspection procedure using “transfer functions” from an existing set of POD curves from a related scenario [15], [16].

Brausch et al. [15] discuss the application of an “inspectability factor” that encapsulates human factors challenges. These factors, which range from 1 to 2, should be used to adjust the expected aNDI (largest crack length that may be assumed to be missed during an inspection).

In summary, as discussed above, there are a number of reasons to doubt the “exact” values for the parameters of a POD curve for any particular application. In addition, it may be useful to conduct “what-if” scenarios to quickly assess the value of changes in the inspection method in reducing the POF or reducing the cost of inspections. As a result, it is useful to have a quantitative estimate how the POF varies as a function of the parameters of a particular POD curve. These questions can be easily estimated once the sensitivities are computed and this information provides an estimate as to how much variation in the POF can be expected due to the mismatch of the environment for development of the POD curve versus its field application. Given the value of sensitivities and the difficulty with current parameter studies and finite difference approaches that require reanalyses, a more formal investigation into sensitivity methods of the POF with respect to the parameters of a POD curve is warranted and presented here.

The paper is organized as follows. The methodology to determine the sensitivities for a single inspection is developed in detail in Section 2 followed by a summary of the results for multiple inspections. Variance estimates of the sensitivities are presented in Section 3. Two numerical examples are presented in Section 4 followed by the Conclusions in Section 5.

Section snippets

Methodology for probability-of-failure sensitivities

The POF for fatigue analysis without inspection is evaluated asPf(t)=g(x,t)0fx(x)dxwhere Pf denotes the POF, t is the time in cycles, x is a vector of random variables, fx(x) is the joint density function of the random variables, and g(x,t) is a limit state function used to define failure. If failure occurs before the observation time, tf0, g(x,t) is less than or equal to zero, i.e., g(x,t) = tf(x)−tf0 where tf is the cycles-to-failure, defined when KI(t) ≥ KIc. KIc is the fracture toughness

Variance estimates

The sensitivity estimates computed using sampling are random variables and their accuracy depends upon the problem under investigation and the number of samples used. An estimate for the variance (or standard deviation) of the sensitivities can be computed be adopting and modifying the variance estimate in [19] asV(Pf/θ)1N2i=1Nq(I(xi,t)Ω(θ,a(yi,tq)))21N(Pf/θ)2The sensitivities follow a normal distribution; therefore, the 95% confidence bounds can be computed asPf/θ1.96VPf/θPf/θ+

Numerical examples

Two examples are presented to demonstrate and verify the methodology. In both cases, the probabilistic sensitivities computed using the equations derived here are compared against finite difference estimates. Finite difference estimates require multiple probabilistic analyses with a separate small perturbation and analysis for each parameter in the POD curves. In addition, the number of samples required to accurately compute small differences in the POF is typically very large; usually an order

Conclusions

NDE methods play a critical role in fracture control plans. The efficacy of the NDE methods is embodied in the POD curve that relates the probability of detecting a defect versus the defect size. A methodology has been developed and demonstrated that will compute the sensitivity of the POF with respect to the parameters of the POD curve. These sensitivities provide a convenient and low cost method to assess the sign and magnitude of potential changes in the POF with respect to variations in the

Acknowledgment

This research was supported by the Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson AFB, under Contract no. FA8650-07-C-5060, Patrick Golden AFRL/RXLMN program manager.

References (19)

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