An up-scaling temperature compensation framework for guided wave–based structural health monitoring in large composite structures

Abstract

Variations in environmental conditions can significantly impair the accuracy and reliability of guided wave structural health monitoring systems. Acquisition of baseline signals over a wide temperature range for the purposes of damage detection and localization is impractical for large composite structures. A novel framework for compensating the effect of temperature at a post-processing stage is presented in this paper to allow updating the compensation factors using observations obtained at different scales. The proposed methodology utilizes observations collected at the lower scales, where a large amount of data under controlled environment is available. Subsequently, the estimated compensation factors are propagated to the higher scales as priors within a Bayesian framework. This way, the measurements required from the high levels are reduced while making it possible to also update the estimated factors during the operation of the structure. The performance of the methodology is evaluated at different scales and compared with the direct use of compensation factors obtained from coupon studies only. It is demonstrated that the proposed methodology improves the fidelity of the compensation algorithm leading to a reduction in the uncertainty of the temperature-compensated signals. Based on the findings of the present study, the reduction in the uncertainty of the compensation improves the performance of both damage detection as well as damage localization in a large composite panel.

Keywords

Guided waves anisotropic layered composites structural health monitoring Bayesian linear regression data-driven temperature compensation

Introduction

In recent years, composite materials are the subject of many research studies in the aviation industry due to their excellent stiffness to weight ratio.^1,2 Composites however are prone to low velocity impact damage that can significantly affect the reliability of the structure and lead to catastrophic failures. Structural health monitoring (SHM) strategies have gained attention as an option for the integrity assessment of a structure. Contrary to traditional non-destructive inspection (NDI) techniques, SHM systems are able to interrogate the structure over long distances, minimize manual labor and human intervention, and allow the transition from the current practices of schedule-based maintenance to a condition-based maintenance philosophy.^3–5 There still are challenges that need to be overcome prior the industrial adoption of SHM in aviation. These challenges can be traced back to the geometrical complexity of aircraft structures, plethora of damage scenarios in composites, cost-benefit limitations, and the wide range of the aircraft environmental and operational conditions (EOC).^6–8

Guided wave–based SHM (GWSHM) systems utilize a network of piezoelectric transducers (PZTs) that can be permanently mounted on or embedded into the structure. Numerous studies have demonstrated that such systems are capable of detecting and localizing barely visible impact damage (BVID) in composite materials.^2,9–13 Typical GWSHM systems are based on the comparison between a baseline measurement that has been taken from a known system state that is considered pristine or defect-free, with a current measurement whose state is unknown. Such systems have been proven effective for complex structures as the baseline and the current measurements already include effects such as geometry and material variability.¹⁴ During this comparison, damage sensitive features are extracted such as time of flight, amplitude, and phase change which provide the basis for damage indexes (DI), for the detection and the localization of defects.^10,15–17

The existence of damage will cause a significant increase in the damage indexes of the sensor pairs in its vicinity. However, non-zero values of $D I_{i}$ might be caused due to intrinsic uncertainties that are unrelated to the existence of damage. To assess the performance of a GWSHM system, all uncertainties pertaining its accuracy must be considered and accounted for. Such uncertainties include electrical noise, external vibrations, different loading conditions, and temperature difference.^13,18,19 Fluctuations in the recorded signals cause deviations from the baseline that can lead to abnormal DIs and erroneous assessments (false alarm or missed-detection).^20–23

Treatment of the variability in the structure’s temperature during the lifetime of an aircraft is a major challenge. Temperature difference between the baseline and a current signal can impair the damage detectability and lead to unacceptably high false positive readings.²⁴ Therefore, it is crucial to adopt a form of temperature compensation to improve the diagnostic reliability of a GWSHM system considering its in-service conditions.²² Various temperature compensation methodologies have been reported in the literature to reduce the effects of temperature difference. The optimal baseline selection (OBS) methodology²³ attempts to find a best match between the current signal and a pre-recorded dataset of baseline signals, taken at different temperatures. Another popular methodology is the baseline signal stretching (BSS) that is contrary to OBS does not require a large database of pre-recorded signals.²⁵ Using BSS, the baseline signal is stretched to match the current signal while the optimal stretch factor is obtained iteratively based on minimization criteria. However, the applicability of the method is limited to a specific temperature range. Combinations of the OBS and BSS approaches have also been proposed to reduce the signals required for the construction of the baseline database considering the BSS temperature range.^24–26 Other attempts follow data-driven routes to achieve temperature compensation. For instance, in Ref. 27, two baseline signals, recorded at different temperatures, are used to reconstruct a baseline at the current temperature while in Ref. 28, a regression model is implemented to describe the phase and amplitude variations in guided waves under different temperatures. In a recent study,²² a regression model was also used to model the temperature variation of the phase and amplitude compensation factors based on measurements on a composite coupon. It was also demonstrated that the compensation factors extracted from a coupon can be applied to more complex structures and mitigate the effects of temperature difference.

Many of these methodologies require the collection of a large dataset of baseline signals or obtaining baseline recordings at specific temperatures. Although this might be possible in a laboratory for simple coupons and smaller components, it can be infeasible for large-scale structures.^22,25 In the aviation industry, the design, development, and validation of a GWSHM system starts at small scale demonstrators (e.g., coupons) and follows the building block approach for aircraft design,²⁹ where at each level, the scale and the complexity of the structure increases. The objectives of the proposed framework are (i) reduce the number of baseline signals required from large-scale components, (ii) account for the structural complexities of the target structure based on measurements taken from the lower scales, (iii) allow for the updating of the estimated compensation factors as more signals become available (e.g., during operation), and (iv) improve the reliability of the temperature compensation method as the structure’s complexity increases. These objectives are desirable attributes for the industrial adoption of GWSHM systems. This study builds upon two main contributions to propose a novel framework for the propagation and the refinement of temperature compensation information moving through the scales. More specifically, in Ref [9], the development of the detection and localization algorithms for large composite structures is presented following the building block approach. Sensor path eligibility criteria are established to ensure that the signals across the different scales are similar. Furthermore, in Ref [22], a data-driven temperature compensation methodology is presented that estimates the compensation factors along different propagation directions using measurements taken from a coupon. In the proposed framework, the temperature compensation factors obtained at the lowest scale (coupon) are used as a starting point and a novel up-scaling methodology is described that introduces additional corrections factors to improve temperature compensation. The purpose of the correction factors is to account for the effect of the structural complexities (such as stringers and different geometry.) that have not been considered. The correction factors introduced are propagated through the scales within a Bayesian regression model.³⁰ The introduction of the Bayesian model allows to use the observations obtained at a previous scale as prior information for the current scale as well as update the current correction factors when new observations become available. Therefore, a rigorous methodology is established that informs the higher scales based on the observations of the lower scales capable of reducing the required number of large-scale experiments required.

The proposed information passing between the different scales is illustrated in Figure 1. A case study is adopted in to demonstrate the methodology. A 1.6 m long flat stiffened panel is considered as the final scale target structure and the aim is to obtain the temperature compensation factors to improve damage detectability and localization for typical aircraft EOC. This scenario presents a case where obtaining baseline measurements from the target structure over a wide range of temperatures might be infeasible. This restriction can be alleviated by exploiting measurements taken from lower scales. Through this case study, the process of transferring observations from the lower scales to the higher ones is exemplified. At the lowest scale (coupon), temperature compensation factors are extracted along different directions. A 0.5 m long monostringer is used as an intermediate scale to introduce geometrical complexities in the estimation of the compensations factors that are found in the flat panel scale but are missing from the coupon scale. Although a single intermediate scale is used here, the methodology is applicable to situations where multiple intermediate scales are needed. All specimens are manufactured using the same layered composite material that exhibits anisotropic behavior. Due to the Bayesian nature of the regression model, it is also possible to update the estimated correction factors within each scale if new measurements become available.

Figure 1.

Propagation of information within the proposed temperature compensation framework.

This contribution is structured as follows: the Methodology section presents the methodology adopted for the proposed framework and contains details of the experiment set-up used, the extraction of the compensation factors, and the Bayesian regression model. In the section Comparison of the Temperature Compensation Methods, the performance of the proposed temperature compensation framework is demonstrated based on measurements obtained at the element and the sub-component level. Then, in the Damage Detectability and Localization Results section, the improvement in terms of defect detectability and localization accuracy is demonstrated when the baseline and the current signals are obtained at different temperatures.

Methodology

Experimental set-up and signal processing

Guided wave measurements are collected from a series of composite structures under different temperatures. All specimens are made of thermoset M21/194/34%/T800S unidirectional prepreg material with stacking sequence ${[\pm 45 / 0_{2} / 90 / 0]}_{s}$ and thickness $t_{lmt} = 2.208 mm$ . DuraAct piezoceramic disks are bonded to the surface of the CFRP using Hexcel Redux 312 adhesive film and the integrity of the bonding is assessed with Electromechanical Impendence Measurements (EMI) by comparing the real part of the impedance between the bonded sensor and the response of a free sensor.³¹ The bonding is deemed healthy when a frequency shift is observed for the first peak. Signal actuation and sensing is carried out using a National Instrument waveform generator and a PXI 5105 Oscilloscope. Narrowband excitation in layered composites is generally preferred due to the dispersive characteristics of guided waves.¹² In this study, a five-cycle Hanning windowed signal is considered that is defined as

V (t) = 0.5 A_{0} [H (t) - H (t - n_{c} / n_{c} f_{c} f_{c})] \sin (2 π f_{c} t) (1 - \cos (2 π f_{c} t / t n_{c} n_{c}))

(1)

where $H (\cdot)$ is the Heaviside function, $n_{c} = 5$ is the number of cycles, and $f_{c}$ is the central frequency. The value $n_{c} = 5$ is selected to generate a narrowband excitation with a short duration. At $f_{c} = 50 kHz$ and $f_{c} = 250 kHz$ , the fundamental wavemodes $A_{0}$ and $S_{0}$ are dominant, respectively.^32,33 Based on the observations in Ref [9], improved damage detection capability is obtained for $f_{c} = 50 kHz$ where the $A_{0}$ wavemode is dominant for the impact events studied. As a demonstrator, an input signal with $f_{c} = 50 kHz$ is used to present the temperature compensation methodology.

To obtain measurements under different EOC, a TVC J2235 environmental chamber is used. Measurements are collected over a range of temperatures, with a step of $5 \circ C$ . Signals are collected from three different scales, namely, the coupon, element, and sub-component scales.³⁴ A summary of the test campaign is reported in Table 1. The geometry and the sensor locations for all specimens considered are illustrated schematically in Figure 2. The sensor numbering indicated in Figure 2 is used throughout this study. It is noted that out of the three flat panels, only FP3 was tested in the environmental chamber while measurements for FP1 and FP2 were collected at room temperature only.

Table 1.

Summary of test campaign for the pristine data collection.

ID	Level	Scale	Bonding process	Temperature range	No. Sensors	No. Signals
C1	Coupon	Coupon	-	$- 45 \circ C \leq T \leq 65 \circ C$	13	600
MS1	Element	Monostringer	Co-Bonded	$10 \circ C \leq T \leq 40 \circ C$	8	70
MS2	Element	Monostringer	Co-Bonded	$10 \circ C \leq T \leq 40 \circ C$	8	210
MS3	Element	Monostringer	Co-Cured	$10 \circ C \leq T \leq 40 \circ C$	8	190
FP1	Sub-Component	Flat Panel	Co-Bonded	$T = 22 \circ C$	24	30
FP2	Sub-Component	Flat Panel	Co-Bonded	$T = 25 \circ C$	24	30
FP3	Sub-Component	Flat Panel	Co-Bonded	$25 \circ C \leq T \leq 50 \circ C$	24	200

Figure 2.

Geometry and sensor locations for a) coupon, b) monostringer, and c) flat panel. Red stars in the flat panel indicate the locations of the impact events.

The behavior of propagating guided waves can be significantly affected by the structural complexities of large structures (e.g., pair distance and number of stiffeners). In such scenarios, the signal might experience excessive attenuation and wavemode conversion making the comparison with lower scales difficult. To allow up-scaling of the compensation algorithm, the sensor pair eligibility criteria set out in Ref [9] are adopted. Eligible pairs must offer additional coverage (i.e., path $1 - 3$ is not selected as it offers the same coverage as paths $1 - 2$ and $2 - 3$ ), share similar attenuation profiles (i.e., similar distances), and cross similar geometric features. These criteria aim at retaining the geometric similarities through the scales, ensuring that the features extracted are transferable. For instance, the sensor groups $1 - 8$ at the monostringer scale is repeating at the flat panel scale (groups $1 - 8$ and $5 - 12$ ). Paths that cross two stringers at the flat panel scale (e.g., $1 - 12$ ) are not considered as they do not appear in the lower scales. The exclusion of these pairs is further justified by the excessive attenuation these pairs exhibit. It is noted here that the proposed framework for temperature compensation can be applied to different sensor network configurations. It is important however that the networks at the different scales share similarities to allow transferring of the observations through the scales.

Let $n_{pairs}$ be the total number of eligible pairs. Then, $P = {p_{i}}, i = 1, \dots, n_{pairs}$ is the set containing all possible pairs and $p_{i} = [p_{i 1}, p_{i 2}]$ is the vector holding the number of the actuator and the sensor. Following the eligibility criteria in Ref [9], $n_{pairs} = 44$ and $n_{pairs} = 82$ are defined for the monostringer and the flat panel, respectively. It is noted that for the coupon case, only pairs starting from PZT 1 were considered and thus $n_{pairs} = 12$ .

Let $r_{i} (t, T)$ denote the signal recorded at temperature $T$ from the $i^{th}$ sensor pair. The signal to noise ratio of each signal is improved by repeating each measurement $10$ times and averaging the signals. A window function is applied to the signal to reduce the effect of boundary reflections, keeping only the first wave packet. Denoting with $d_{i}$ the distance between the sensors of a pair and with $c_{i}$ the group velocity of a pair the effective signal is obtained as

s_{i} (t, T) = r_{i} (t, T) W_{i} (t)

(2)

where $W_{i} (t) = H (t - t_{a}) - H (t - t_{b})$ is a window function and $t_{a} = d_{i} / d_{i} c_{i} c_{i}$ and $t_{b} = t_{a} + 1.1 \cdot n_{c} / n_{c} f_{c} f_{c}$ correspond to the arrival time and duration of the wave packet. The factor 1.1 has been introduced in the expression of $t_{b}$ to make sure that the entire first wave packet is included within the time window, accounting for dispersion and the influence of the stiffener. The definition of the time window requires the estimation of $t_{a}$ and $t_{b}$ that can be influenced by uncertainties such as material variability, signal dispersion, and boundary reflections. These can influence damage detectability and localization as well as the estimation of the compensation factors.

Damage detection and localization can be carried out by comparing two signals: the first signal is a reference signal $s_{i}^{ref} (t, T)$ that is recorded at the pristine and defect-free state while the second is taken from the current state of the structure (which can be either pristine or damaged).^13,14,17 In this study, a damaged index based on the correlation coefficient is used computed as

D I_{i} = 1 - \frac{cov (s_{i}^{ref} (t, T), s_{i} (t, T))}{σ (s_{i}^{ref} (t, T)) σ (s_{i} (t, T))}

(3)

where $cov (x, y)$ is the covariance between signals $x$ and $y$ while $σ (x)$ is the standard deviation. A different damage index could also be used without affecting the proposed temperature compensation framework.

Due to the anisotropic behavior of the composite material, the wave propagation direction influences the characteristics of the signal. The propagation direction, $ϕ_{i}$ , is defined based on the location of the actuator and the sensor for each pair. The compensation factors and the group velocity are therefore dependent on the propagation direction. To account for this effect, measurements are taken along multiple directions starting at the coupon scale, as illustrated in Figure 2.

Temperature compensation and extraction of correction factors

The effect of temperature difference on the recorded guided wave signals can be approximated using the following expression^22,35

{\hat{s}}_{i} (t, T_{0}) = α (ϕ, T_{1}) s_{i} (β (ϕ, T_{1}) t, T_{1})

(4)

where $T_{0}$ is the target temperature and $α and β$ are two compensation factors that account for the temperature effect on the amplitude and the phase of the signal, respectively. The values of $α and β$ are influenced by the temperature dependency of the composite elastic properties, the piezoelectric properties of the sensors as well as the bonding medium.³⁶ Furthermore, due to the anisotropic behavior of the composite considered, the compensation factors depend on both the temperature and the propagation direction $ϕ_{i}$ .²² Using equation (4), it is possible to approximate the signal at a new temperature, $T_{0}$ , if it is measured at a different temperature, $T_{1}$ . Since it is infeasible in SHM application to collect baseline signals at all possible temperatures within the operational range of an aircraft, the adverse effects of temperature difference between the baseline signal and the current signal can be reduced by compensating the current measurement.

In Ref [22], the temperature compensation factors were estimated based on coupon measurements following a data-driven approach. It was demonstrated that these factors can be treated as material properties and although computed from simple coupons, can be used to mitigate the effects of temperature difference at complex structures. The efficiency of the compensation can be further improved by accounting for structural complexities (e.g., stringers) of the host structure. The coupon scale compensation factors are taken from Ref [22] directly while this study focuses on applying corrections to higher scales.

The temperature compensation factors can be written as

α (ϕ, T) = α^{coupon} (ϕ, T) + α^{corr} (ϕ, T)

(5a)

β (ϕ, T) = β^{coupon} (ϕ, T) + β^{corr} (ϕ, T)

(5b)

where $α^{coupon} (ϕ, T)$ and $β^{coupon} (ϕ, T)$ are the compensation factors computed from the coupon scale while $α^{corr} (ϕ, T)$ and $β^{corr} (ϕ, T)$ are the correction factors that account for the structural differences of each scale.

Due to the complex interaction of temperature with the wave propagation characteristics, a data-driven approach based on experimental measurements taken at different temperatures is implemented to estimate the optimum $α$ and $β$ factors at each scale. Using equation (4), the following fitness function between a compensated signal and a true signal can be established

f (θ_{i}) = \sum {[{\hat{s}}_{i} (t, T_{0}) - s_{i} (t, T_{0})]}^{2} = \sum {[a (ϕ_{i}, T_{1}) s_{i} (β (ϕ_{i}, T_{1}) t, T_{1}) - s_{i} (t, T_{0})]}^{2}

(6)

where $θ_{i} = [a (ϕ_{i}, T), β (ϕ_{i}, T)]$ is the vector of compensation factors needed to compensate a signal from the current temperature, $T_{1},$ to the target temperature, $T_{0}$ . The optimum compensation factors can be estimated by minimizing the fitness function

{\hat{θ}}_{i} = \arg min_{α, β} f (θ_{i})

(7)

Based on the results presented in Ref [22], the constrains $α \in [0.4, 1.6]$ and $β \in [0.8, 1.2]$ are used for the optimization problem since they cover the expected values for temperature difference up to $| Δ T | = 50 \circ C$ for the coupon scale. After estimating the optimum $α$ and $β$ factors for a given scale, equations (5a) and (5b) can be used to estimate the correction factors as

a^{corr} (ϕ_{i}, T) = a (ϕ_{i}, T) - a^{coupon} (ϕ_{i}, T)

(8a)

β^{corr} (ϕ_{i}, T) = β (ϕ_{i}, T) - β^{coupon} (ϕ_{i}, T)

(8b)

As an example, the temperature compensation for two different sensor pairs of FP3 is illustrated in Figure 3. A signal that is measured at $T_{1} = 35 \circ C$ is compensated to $T_{0} = 25 \circ C$ using the compensation factors obtained from the coupons as well as by minimizing equation (7). The true response at $T_{0} = 25 \circ C$ is also included for validation. Using the coupon, compensation factors directly perform better for pair 6–5 compared to pair 6–2, as for pair 6–5, both PZTs are in the same bay and in close proximity. The corrected factors improve the compensation for both sensor pairs as the compensated signals match better the true response.

Figure 3.

Comparison of the compensation performance when the coupon and the corrected compensation factors are used along propagation directions $ϕ = 0^{o}$ (pairs 6–5) and $ϕ = 90^{o}$ (pairs 6–2).

To evaluate the effect of uncertainty in the estimation of the time window (see equation (2)), on the estimation of the compensation factors, the arrival time $t_{a}$ is modified as $t_{a} = k \cdot d_{i} / d_{i} c_{i} c_{i}$ , where $k \in [0.8, 1.2]$ . Therefore, a ± 20% variation is allowed in the time of arrival estimation. Factors $α (ϕ, T)$ and $β (ϕ, T)$ estimated using equation (7) are compared and the absolute percent error is illustrated in Figure 4. For both factors, pair 6–2 that crosses the stringer exhibits a higher variation compared to 6–5; however, the influence of the time window modification is below 0.8% and 0.05% for $α (ϕ, T)$ and $β (ϕ, T)$ , respectively.

Figure 4.

Estimated $α (ϕ, T)$ and $β (ϕ, T)$ error along flat panel pairs 6–5 and 6–2 for variations in the time window considered.

Bayesian regression model with radial basis functions

The use of the Bayesian inference can provide the mathematical foundations for refining prior information and accounting for uncertainties.³⁷ An impediment of the common compensation approaches is the difficulty to collect the large number of data required in real large-scale structures. The introduction of a Bayesian regression model in the proposed framework allows to propagate the information from the lower to the higher scales in the form of priors as well update the correction factors as new observations become available (e.g., during the operation of the aircraft).

The correction factors are modeled using a linear regression approximation³⁰

y (x) = \sum_{i = 1}^{n} (w_{i} ψ_{i} (x) + e_{i}) = w^{T} φ (x) + e

(9)

where $w = {(w_{1}, w_{2}, \dots, w_{N})}^{T}$ is a $N \times 1$ vector of weights, $ψ (x) = {(ψ_{1} (x), ψ_{2} (x), \dots, ψ_{N} (x))}^{T}$ is the $N \times 1$ vector of basis functions, $x$ is the input, and $e ~ N (0, σ_{e})$ is the noise in the model. Incorporating nonlinear basis functions in equation (9), it is possible to capture complex model responses.^38,39 Here, the radial basis function (RBF) is chosen and expressed as³⁰

ψ_{i} (x) = \exp (- \frac{x - μ_{i}}{2 s})

(10)

where $s$ and $μ_{i}$ denote the length scale and the location of the RBF. Incorporating the RBF in the regression model for the temperature compensation introduces an intrinsic length scale that will limit the influence of each observation to its vicinity only. The RBF is chosen here due to the smooth variation of the temperature compensation factors that has been observed experimentally.

For the Bayesian treatment of the linear regression model, conjugate priors are selected that allow all expressions to be solved in closed form.⁴⁰ It is assumed that both the weights $w$ and the model variance $σ_{e}^{2}$ are unknown parameters that need to be inferred. A common parameterization of the problem is the use of a Normal Inverse Gamma (NIG) prior³⁸ and can be written as

p (w, σ_{e}^{2}) = p (w | σ_{e}^{2}) p (σ_{e}^{2}) ~ N (w | μ, σ_{e}^{2} V) IG (σ_{e}^{2} | a, b) = NIG (μ, V, a, b)

(11)

where $μ$ is the prior mean of the weights, $V$ is the prior covariance matrix, and $IG (a, b)$ denotes the inverse gamma prior distribution for $σ_{e}^{2}$ ⁴⁰ that is given as

p (σ_{e}^{2}) = \frac{b^{a}}{Γ (a)} {(\frac{1}{σ_{e}^{2}})}^{a + 1} \exp (- \frac{b}{σ_{e}^{2}})

(12)

where $Γ (x)$ denotes the gamma function with shape $a$ and rate $b$ .

For a dataset of $M$ observations $Y = {[y_{i}]}_{i = 1}^{M}$ associated with inputs $X = {[x_{i}]}_{i = 1}^{M}$ , the likelihood is^38,41

p (Y | w, σ_{e}^{2}) = Π_{i = 1}^{M} N (y_{i} | w^{T} φ (x_{i}), σ_{e}^{2})

(13)

Since the NIG prior is a conjugate family for the regression problem, the posterior is also NIG distributed as

p (w, σ_{e}^{2} | Y) = NIG (μ^{*}, V^{*}, a^{*}, b^{*})

(14)

where $μ^{*}, V^{*}, a^{*}, and b^{*}$ are the posterior parameters computed as³⁸

V^{*} = {(V^{- 1} + Φ^{T} Φ)}^{- 1}

(15a)

μ^{*} = V^{*} (V_{0}^{- 1} μ + Φ^{T} y)

(15b)

a^{*} = a + M / M 2 2

(15c)

b^{*} = b + 0.5 (μ^{T} V^{- 1} μ + Φ^{T} Φ - μ^{* T} V^{* - 1} μ^{*})

(15d)

where $Φ$ is the $M \times N$ design matrix³⁰ and its elements are defined as $Φ_{i, j} = φ_{j} (x_{i})$ .

Predictions at a new set of inputs $\hat{X}$ can be obtained through the posterior predictive distribution given as³⁸

p (\hat{Y} | \hat{X}, Y) = MVT (\hat{Y} | {\hat{X}}^{T} μ^{*}, \frac{b^{*}}{a^{*}} (I + \hat{X} V^{*} {\hat{X}}^{T}), 2 a^{*})

(16)

where $MVT$ denotes the multivariate T distribution and $I$ is the unit matrix.

Damage detection and localization method

Adoption of an appropriate methodology and the selection of a threshold value are required to reliably distinguish between pristine and damaged signals. In this study, the methodology presented in Ref [9] and briefly described here is adopted to demonstrate potential improvements in terms of damage detectability when the proposed temperature compensation framework is employed. This detection method is selected as it follows the sensor adaptation, allowing the transition between different scales.

The damage detection methodology is based on computing $μ_{DI}$ and $σ_{DI}$ as

μ_{DI} = \frac{\sum_{1}^{n_{pairs}} D I_{i}}{n_{pairs}}

(17a)

σ_{DI} = \sqrt{\frac{\sum_{1}^{n_{pairs}} {(D I_{i} - μ_{DI})}^{2}}{n_{pairs} - 1}}

(17b)

The values computed from equations (17a) and (17b) are stored in the vector $D_{i} = [μ_{DI, i}, σ_{DI, i}]$ . If $N$ measurements are available, the damage vector is constructed as $D = {[D_{1}^{T}, D_{2}^{T}, \dots, D_{N}^{T}]}^{T}$ . The current state of the specimen is then identified using an outlier detection algorithm. For the outlier analysis, $D$ is compared with a reference sample of pristine measurements, denoted as $D^{*}$ . Thus, if the discrepancy between $D$ and $D^{*}$ in the selected feature space is above a predefined threshold value, the specimen is identified as damaged. A common and versatile outlier analysis tool for structural health monitoring applications is the Mahalanobis distance (MD).^9,42–44 The MD for a measurement is computed as

M D_{i} = \sqrt{{(D_{i} - μ^{*})}^{T} S^{- 1} (D_{i} - μ^{*})}

(18)

where $S = cov (D^{*})$ and $μ^{*}$ is the mean vector of $D^{*}$ . The performance of the outlier analysis however depends on the pristine data collected. As stated in Ref [9], the pristine data are affected by uncertainties in the EOC. Temperature compensation can reduce the effects of temperature difference, improving the false positive and false negative indications of the system. The proposed temperature compensation scheme can reduce the number of measurements required from the full-scale structure and improve the efficacy of the compensation methodology.

Damage localization is also an integral part of a GWSHM system as it can provide valuable information regarding the damage location. The damage localization algorithm is executed after the damage detection has indicated the existence of damage. Details on the localization imaging used are included in Appendix A.

Comparison of the temperature compensation methods

One of the drawbacks of the data-driven methods is the requirement of multiple measurements to estimate the compensation factors. The capability of the proposed framework to propagate and improve the accuracy of the temperature compensation algorithm is demonstrated in this section. For comparison between different compensation options, the following cases are presented:

Case 1: No compensation is applied to the signals. This case is included for benchmarking purposes.

Case 2: Temperature compensation I carried out using only the factors extracted from the coupon data. Information is passed from the coupon scale to the current scale directly.

Case 3: The correction factors computed from the previous scales are also considered. In this setting, information is passed from the lower to the higher scales. Case 2 is the starting point for Case 3.

Case 4: The correction factors that have been passed on by the lower scales are updated in the current scale as new observations become available. Thus, Case 3 provides the starting point for Case 4.

Monostringer scale

As a first example, the measurements collected from the monostringers are used to demonstrate the potential performance gain of introducing the correction factors for the temperature compensation. In this example, a baseline pristine signal, measured at $T_{0} = 25 \circ C$ is used for each pair as $s_{i}^{ref} (t, T)$ in equation (3) and results are presented for Cases 1, 2, and 3. The factors $a^{coupon}$ and $β^{coupon}$ are extracted from the coupon²² along multiple propagation directions $0 \leq ϕ \leq 2 π$ . Next, it is assumed that data at different temperatures are available from specimen MS3 and will be used for the extraction of the corrections factors that will be applied for specimens MS1 and MS2. This specific choice is made because for MS3, the skin and the stringer are bonded following a Co-Curing process while for MS1 and MS2, the Co-Bonding process is followed. This demonstrates the applicability of the method when used for specimens in the same scale that share the same material and geometry but have slight manufacturing differences. Raw signals from a sensor pair that crosses the stringer are compared in Figure 5 for a Co-cured and a Co-bonded monostringer. A schematic illustration of the information passing is depicted in Figure 6.

Figure 5.

Comparison of signals between a co-bonded (MS2) and a co-cured (MS3) monostringer. The path illustrated crosses the stringer at $90 \circ$ .

Figure 6.

Illustration of the data passing route in the monostringer example.

Using the signals collected for MS3, the optimum compensation factors $a (ϕ, T)$ and $β (ϕ, T)$ are estimated at different temperatures by solving equation (7) and subsequently, the correction factors are computed using the coupon factors through equations (8a) and (8b). The Bayesian Regression model is then fitted to the data through the computation of the posteriors in equations (15a)–(15d). It is assumed that no prior information is available for the correction factors of the monostringer and therefore the uninformative priors $μ = 0, V = I, a = 0$ , and $b = 0$ have been used.³⁸ The correction factors computed for specimen MS3 and the fitted regression model are presented in Figure 7 for $ϕ = 0 \circ$ (pairs $1 - 2, 2 - 3, 3 - 4, 5 - 6, 6 - 7$ , and $7 - 8$ ), $ϕ = 90 \circ$ (pairs $5 - 1, 6 - 2, 7 - 3, and 8 - 4$ ), and $ϕ = 125.82 \circ$ (pairs $6 - 1, 7 - 2, and 8 - 3$ ). These propagation directions are selected to demonstrate situations where the sensor pair path does not cross the stringer, the path is perpendicular to the stringer, and the path crosses the stringer at an angle.

Figure 7.

Fitted regression model for the correction factors $a^{corr} (ϕ, T)$ and $β^{corr} (ϕ, T)$ computed based on the signals collected from MS3 monostringer.

The use of the RBF in the Bayesian regression model allows to capture nonlinearities in the response of the correction factors $a^{corr} (ϕ, T)$ and $β^{corr} (ϕ, T)$ along different propagation directions. Furthermore, the introduction of the length scale to the regression model makes the posterior return to 0 away from the observed temperatures. This is beneficial because if a polynomial regression is used, it could lead to erroneous estimations of the compensation factors if only a small dataset is available, or extrapolation is attempted away from the observed temperatures. Using the proposed regression in such cases, the compensation factors return to the coupon only values which are treated as the best estimate if no other data are available.

For comparison, the optimum compensation factors are also computed for specimens MS1 and MS2. The results are reported in Figure 8. In the same figure, the coupon (dashed line) and the corrected compensation factors after observing the measurements from MS3 (solid line) are also reported. It is illustrated that after observing the measurements from MS3, the corrected compensation factors are closer to the optimum values computed using equation (7) for MS1 and MS2, compared to using the compensation factors from the coupon scale directly.

Figure 8.

Comparison of compensation factors for different monostringer data.

Despite MS3 being Co-Cured, refining the compensation factors for specimens with similar geometry can improve the temperature compensation algorithm and help further mitigate the influence of temperature difference. The Bayesian regression model adopted here allows to continue the updating the correction factors after observing the MS3 data. For instance, the correction factors estimated from MS3 could be updated with the observations from MS1 before being applied to MS2.

The temperature difference between the baseline signal and the current leads to the computation of a non-zero damage index that could potentially mask the existence of damage in the structure. In Figure 9, the damage indexes for two signals of MS1 with temperature difference $Δ T = 10 \circ C$ and $Δ T = 15 \circ C$ from the baseline are reported. The signals used are collected from the pristine state of the structure. Therefore, if the temperature compensation was perfect, the damage indexes should be close to zero along all paths. Compared to not applying any compensation (Case 1), Case 2 is capable of reducing the computed damage indexes along all sensor pairs. Incorporating the estimated correction factors predicted from the Bayesian regression model (Case 3), it is possible to further improve the temperature compensation process reducing the residuals. It is noted that the mean estimates of the correction factors from the MS3 data have been used for the temperature compensation in Case 3.

Figure 9.

Comparison of the computed damage indexes from pristine signals collected from MS1 when $Δ T = 10 \circ C$ (top) and $Δ T = 15 \circ C$ (bottom). The y-axis is plotted in logarithmic scale.

The value of $μ_{DI}$ (equation (17a)) is used as a metric to evaluate the improvement after incorporating the correction factors. A signal at $T = 25 \circ C$ is used as a baseline and $μ_{DI}$ is computed for all measurements collected from MS1 and MS2. The selected baseline temperature leads to a maximum temperature difference of $| Δ T | = 15 \circ C$ (see also Table 1). All signals are collected from the pristine state and therefore any residuals observed are caused due to the temperature difference and the measurement noise. The samples collected from MS1 and MS2 (Table 1) are arranged based on their temperature difference from the baseline and the results are reported in Figure 10. At $Δ T = 0 \circ C$ , there is no need for temperature compensation and only the signal noise is observed. As $Δ T$ increases, the need for temperature compensation becomes evident and $μ_{DI}$ increases. At all temperatures, the addition of the correction factors computed from MS3 improves the temperature compensation reducing the computed residuals. Using Case 1 as a benchmark, the relative reduction of the residuals is summarized in Table 2. On average, the proposed addition of the correction factors can lead to an improvement of 5.41% in the temperature range considered.

Figure 10.

Comparison of the different compensation approaches for the available monitoring history. The y-axis is plotted in logarithmic scale.

Table 2.

Comparison in the relative reduction of $μ_{DI}$ at different temperatures when Cases 2 and 3 are used for the monostringer data.

	$Δ T = - 15 \circ C$	$Δ T = - 10 \circ C$	$Δ T = - 5 \circ C$	$Δ T = 0 \circ C$	$Δ T = 5 \circ C$	$Δ T = 10 \circ C$	$Δ T = 15 \circ C$
Case 2	86.42%	84.80%	79.89%	0.001%	75.03%	79.56%	79.30%
Case 3	91.17%	89.70%	84.37%	0.012%	79.96%	85.92%	86.40%
Gain	4.75%	4.9%	4.48%	0.011%	4.93%	6.35%	7.10%

Flat panel scale

In the previous section, the proposed temperature compensation was tested for specimens that belong to the same scale to validate the introduction of the correction factors. In this section, the proposed methodology is applied to the measurements taken from FP3 to demonstrate a case where the correction factors computed at a lower scale are transferred to the higher scales.

Cases 1 and 2 are the same as described in the previous example, corresponding to the no compensation and coupon compensation only. In Case 3, the correction factors are computed at the monostringer scale. Because the flat panels are also Co-Bonded, measurements taken from MS1 and MS2 are used for the estimation of the correction factors. Lastly, in Case 4, updating of the correction factors is allowed (see Figure 11). Let $k$ denote the current scale. If information is available from the previous scale, the prior mean of the correction factors in the Bayesian regression model is set as $μ^{k} = μ^{k - 1}$ . This allows information to be passed across the levels. Since at the beginning of the iterations no data have been observed at the current scale, then the rest of the priors are set to $V = I, a = 0$ , and $b = 0$ to be uninformative.

Figure 11.

Propagation of the temperature compensation factors from the coupon to the sub-component level.

Such situation can arise in practice in the case of large structures. It can be impractical to collect data at wide range of temperatures in the higher scales. Thus, using information from the lower scales provide a starting point for the estimations. Then, as new data become available during the operation of the structure, the estimations are updated. Each time new observations are available, the posteriors become the new priors and the Bayesian inference is repeated. In this example, $μ^{k - 1}$ denotes the posterior mean of the correction factors from the monostringers while $μ^{k}$ the prior mean of the correction factors at the flat panel level.

The measurements from FP3 with temperature range $25 \circ C \leq T \leq 50 \circ C$ are used to evaluate the performance of the proposed method. Similar to the monostringer example in the previous section, the baseline signals have been recorded at $T_{0} = 25 \circ C$ . In this example, the analysis is repeated multiple times as the updating method in Case 4 can be influenced by the order the signals are received. In total, the analysis is repeated $n_{total} = 20$ times and each time $μ_{DI}$ is computed. At each repetition, the measurements are shuffled randomly creating a new possible permutation of received measurements. Each time a signal is received, it is compensated based on the currently available data and subsequently, the regression model is updated to improve the fidelity of the estimations.

The updating process of $β^{corr}$ is illustrated schematically in Figure 12 for a random permutation. Orange squares indicate data received at the current iteration while gray squares indicate data that have already been considered during the updating phase of the methodology. At iteration 2 (top left in Figure 12), the correction factors for Case 4 have been updated for the observations at $35 \circ C$ which has improved slightly the prediction for $40 \circ C$ . Cases 3 and 4 deviate when the current measurement’s temperature is $45 \circ C$ and $50 \circ C$ (top right and bottom left in Figure 12, respectively). This is expected as the temperature range of the monostringer data is $10 \circ C \leq T \leq 40 \circ C$ and the regression model returns to the coupon only case outside the range of observed temperatures. After many different temperature samples have been used for the updating of Case 4 (bottom right in Figure 12), the mean estimations of Case 3 and Case 4 are very close in the temperature range $25 \circ C \leq T \leq 40 \circ C$ . Therefore, even if no data is available from the current scale, observations from the lower scales can be used to improve the performance of the temperature compensation method. This has many industrial applications as in the lower scales, specimens are cheaper to manufacture and collecting larger sets of data using environmental chambers is more manageable.

Figure 12.

Illustration for the dynamic updating of $β^{corr} (90^{o}, T)$ within the Bayesian framework as new data become available.

The computed $μ_{DI}$ for all 20 permutations are grouped according to the measurement temperature and reported in the Figure 13 box plot. Cases 1, 2, and 3 are not influenced by the permutations of the signals and thus lead to a very small scatter. At $T = 25 \circ C$ , the results from all cases are similar and are mainly affected by uncertainties such as external noise. Case 3 outperforms Case 2 at all temperatures, however, its influence is reduced at $45 \circ C$ while at $50 \circ C$ , the two cases almost coincide.

Figure 13.

Comparison of the estimated $μ_{DI}$ at different temperatures. Y-axis is plotted in logarithmic scale.

Case 4 indicates outliers that are influenced by the arrival sequence of the signals. These outliers indicate initially a similar response to Case 3. This is expected as Case 3 was used as starting prior for the correction factors. As the regression model gets updated, the performance of the compensation scheme improves. Take $T = 45 \circ C$ as an example. Although initially the $μ_{DI}$ values close to Case 3 are observed, its value converges to around 0.02 after all update permutations. Using $μ_{DI}$ of Case 1 as a benchmark, the relative reduction in $μ_{DI}$ for all other cases is reported in Table 3. Compared to using the coupon compensation factors only (Case 2), incorporation of the monostringer data (Case 3) can lead to an average $μ_{DI}$ reduction of 7.19% across the temperature range considered. If updating of the correction factors is also employed as new observations become available (Case 4), the reduction can be increased up to 12.49%. Furthermore, the improvement in the temperature compensation can be demonstrated by computing the mean absolute percentage error (MAPE) between the mean predicted compensation factors and the observed values after the updating process has completed (see Figure 12). For instance, for $β (ϕ = 90^{o}, T = 45 \circ C)$ , the MAPE is computed as $4.8 \cdot 10^{- 3}$ , $3 \cdot 10^{- 3}$ , and $0.6 \cdot 10^{- 3}$ for Cases 2, 3, and 4, respectively. Therefore, if all monostringer observations are available to the framework, the MAPE can be reduced by 37.5% while if the flat panel measurements are provided, the reduction is 87.5%.

Table 3.

Relative reduction of $μ_{DI}$ when MS1 and MS2 data are used as a starting point for the flat panel correction factors.

	$Δ T = 1 \circ C$	$Δ T = 5 \circ C$	$Δ T = 10 \circ C$	$Δ T = 15 \circ C$	$Δ T = 20 \circ C$	$Δ T = 25 \circ C$
Case 2	27.83%	73.69%	80.89%	81.24%	81.14%	79.74%
Case 3	31.75%	83.87%	91.94%	93.03%	86.70%	80.39%
Case 4	31.86%	89.12%	94.62%	95.67%	94.92%	93.26%

The updating process could entail the risk of rendering the system insensitive to damage progression. In this scenario, a composite structure with only four bays has been considered. The value of the proposed framework however lies in its applicability to large-scale structures. The existence of damage will only affect the sensor readings in its immediate vicinity. Furthermore, measurements from similar aircraft in the fleet would also contribute to the database. It is therefore expected that the cases where undetected damage is included in the updating process will contribute only to a small fraction of the available data. Modifications in the regression method implemented could be explored to improve robustness to outliers.⁴⁵

As a final example, the analysis is repeated with the difference that instead of having measurements available from specimens MS1 and MS2, only measurements from specimen MS3 are available. This is the worst-case scenario where the available measurements from the lower scales belong to a specimen with a different bonding process compared to the target structure (flat panel). The resulting reduction in $μ_{DI}$ for this scenario is summarized in Table 4. The computed values are in very close agreement with the results obtained when the MS1 and MS2 data were used (Table 3). Therefore, the applicability of the methodology is not affected significantly between Co-bonded and Co-cured structures and information passing though the scales can be achieved. This highlights that the most important features in predicting the propagation behavior of the guided waves are the material, composite layup, and the geometrical complexities which are used in this research to up-scale the proposed temperature compensation methodology.

Table 4.

Relative reduction of $μ_{DI}$ when MS3 data are used as a starting point for the flat panel correction factors.

	$Δ T = 1 \circ C$	$Δ T = 5 \circ C$	$Δ T = 10 \circ C$	$Δ T = 15 \circ C$	$Δ T = 20 \circ C$	$Δ T = 25 \circ C$
Case 2	27.8%	73.69%	80.89%	81.24%	81.14%	79.74%
Case 3	32.36%	84.80%	91.87%	93.08%	86.78%	80.39%
Case 4	32.63%	89.52%	94.55%	95.68%	95.01%	92.74%

Damage detectability and localization results

Damage detection

The motivation for the development of an accurate and robust temperature compensation algorithm is driven by the requirement for improved damage detection capabilities. To evaluate damage detectability when the proposed framework is employed, measurements are collected from the flat panels after a series of impacts. In total, eight impact events are carried out with energies between $20 J$ and $35 J$ . Four of the impacts were located at the foot of the stringer, two between the feet and two in a bay. The impact campaign is summarized in Table 5 and the impact locations are indicated with red stars in Figure 2. All measurements were collected at room temperature around $25 \circ C$ . The impact sequence for each panel follows the assigned impact ID order.

Table 5.

Summary of impact tests performed on the flat panels.

Panel ID	Impact ID	X $[mm]$	Y $[mm]$	Energy $[J]$	BVID $[m m^{2}]$	Disbond $[m m^{2}]$	$T [\circ C]$
FP1	1	394	−90	20+25+35	373	1638	22
FP2	1	394	−90	30	392	1421	25
FP3	1	−362	127	20	611	−	26
	2	−392	−163	35	471	2146	26
	3	−180	−50	20	320	−	25
	4	362	127	20	543	−	26
	5	392	−163	35	610	2697	27
	6	268	−41	20	271	−	25

For the damage detection, it is first required to construct the reference pristine vector $D^{*}$ . To capture the temperature uncertainty in the definition of $D^{*}$ , it is assumed that the baseline signal temperature is $T_{0} = 25 \circ C$ and pristine signals in $25 \circ C \leq T \leq 40 \circ C$ are used in equations (3), (17a), and (17b). The temperature difference between the baseline and the current pristine signals is compensated with the compensation cases presented in the previous section and the results are plotted in Figure 14. In all cases, as the temperature increases, the relationship between $μ_{DI}$ and $σ_{DI}$ varies almost linearly. Compared to Case 2, Cases 3 and 4 significantly reduce the spread as the temperature compensation scheme becomes more accurate. Cases 3 and 4 lead to similar results as for the temperature range considered, the monostringer correction factors are very close to the factors after the update.

Figure 14.

Illustration of the threshold limit when the temperature difference is up to $25 \circ C$ using different compensation methodologies. Red dashed line indicates the iso-contour with $4.29$ that is used as a threshold.

Distinction between a pristine and a damaged signal is done by setting up a threshold value, $Th$ . The threshold value is set by selecting the probability that a pristine measurement is misclassified as damaged. In this study, the probability is set at 99.9% to reduce the false positive indications. The threshold value is computed from a chi-squared distribution with $n_{dofs} = 2$ degrees of freedom^9,46 leading to $Th = 4.29$ . In Figure 14, the points that lie on the ellipsoid defined by $Th$ are indicated with a red dashed line. Essentially, the threshold specifies a distance above which a new measurement is considered as damaged. This distance is expressed in standard deviations, from the normal distribution that is estimated using the pristine data.

In Figure 15, the damage vector $D$ is reported that contains the $[μ_{DI}, σ_{DI}]$ pairs from the signals recorded after all impact events on the flat panels. For each impact case, the values $μ_{DI}$ and $σ_{DI}$ vary based on the severity and the location of the damage. Impacts 1 and 4 from panel FP3 are the closest events to the pristine data. This is because these impacts are located between the feet of the stringers, making damage detection difficult. On the other hand, impacts FP1-1, FP2-1, FP3-2, and FP3-5 are located at the stiffener bond-line and caused disbond between the skin and the stringer (Table 5), resulting in the highest $μ_{DI}$ and $σ_{DI}$ values. Impacts FP3-3 and FP3-6 are located in the bay, at close proximity to the sensors and are also easily detectable. Furthermore, it can be observed that for the impacted data, the ratio $σ_{DI} / σ_{DI} μ_{DI} μ_{DI}$ is increased. This is because the impact damage affects mainly the sensor pairs in its vicinity producing higher values of $σ_{DI}$ compared to a temperature increase.

Figure 15.

Illustration of damage vector $D$ and detection results for Cases 2 and 3.

Apart from impact 1 of FP3, all other events are correctly identified, irrespective of the compensation method implemented. For impact 1 of FP3, when Case 2 is used, the detection methodology produced six False Negative (FN) indications. Transitioning to Case 3, the uncertainty in the temperature compensation is decreased and the number of false positive indications are reduced to 1 (see also Figure 15). A single false negative is also reported when case 4 is implemented for the temperature compensation.

Since the uncertainty in the pristine data is reduced, the threshold distance defines a smaller region where pristine data are expected (see red dashed line in Figures 14 and 15). As such, when the temperature compensation scheme is improved, the Mahalanobis distance between $D$ and $D^{*}$ is increased since the uncertainty in the pristine data is reduced. The mean distance $(μ_{MD})$ for each impact event is summarized in Table 6. Compared to Case 2, Cases 3 and 4 indicate higher detectability for all impact events. Case 3 specifically, leads to a 14.45% higher distance estimation for FP3-1, which is the closest event to the pristine data. If the correction factors are also updated as new observations become available, it is estimated that the distance can be increased up to 23.57%. The increase in the Mahalanobis distance reflects an increase in the reliability of the outlier analysis as the impact data are further apart from the pristine ones in the 2D feature space implemented here.

Table 6.

Comparison of the mean distance between the pristine $(D^{*})$ and the current $(D)$ damage vectors.

	Case 2	Case 3		Case 4
Impact ID	$μ_{MD}$	$μ_{MD}$	% gain	$μ_{MD}$	% gain
FP1 – 1	32.56	36.12	10.93	38.61	18.57
FP2 – 1	20.90	22.23	6.39	23.27	11.37
FP3 – 1	6.39	7.31	14.45	7.90	23.57
FP3 – 2	44.19	47.67	7.84	50.27	13.76
FP3 – 3	16.07	17.28	7.51	18.17	13.03
FP3 – 4	12.00	13.38	11.50	14.31	19.24
FP3 – 5	47.73	50.27	5.32	52.43	9.84
FP3 – 6	26.13	28.62	9.54	30.39	16.32

Damage localization

The impact sequence FP3-1 to FP3-6 is used to study damage localization with multiple impact events and highlight the complexity of the scenario assumed. To also consider the effect of temperature difference, the baseline signal temperature is selected as $T_{0} = 35 \circ C$ leading to a temperature difference of approximately $Δ T = 10 \circ C$ with the impacted measurements. The imaging results for impacts FP3-1 till FP3-3 are plotted in Figure 16 following the sequence they were conducted, with brighter colors indicating the possible location of the damage. The region around the 12 sensors is discretized into a $100 \times 100$ grid of pixels, leading to a $4.3 \times 6.4 mm$ pixel size which is adequate to resolve spatially the size of the damages considered. In all compensation cases, the location of the first impact event, FP3-1, is successfully localized when it is first introduced. After events FP3-2 and FP3-3, the location of FP3-1 is not evident. This is because FP3-1 is located between the feet of the stringer, leading to lower damage indexes compared to FP3-2 and FP3-3. Nevertheless, Cases 3 and 4 lead to lower values away from the impact events and provide a smaller region of high values near the true impact locations, increasing the reliability of the localization algorithm.

Figure 16.

Localization imaging results for a sequence of impacts with $Δ T = 10 \circ C$ temperature difference. The impact sequence follows from top to bottom.

To evaluate the localization for each compensation method, the accuracy and the precision of the estimated locations are computed following BS ISO 5725-1.^47,33 Let $x_{d} = (x_{d}, y_{d})$ denote the true location of the damage, $x_{e} = (x_{e, i}, y_{e, i})$ is the vector containing the estimated damage location based on the available monitoring history, and $μ_{e} = ({\bar{x}}_{e}, {\bar{y}}_{e})$ is the mean estimate of the location. The accuracy of the estimate can then be computed as the Euclidian distance between the estimated and the true location as

Acc = \sqrt{{(x_{d} - {\bar{x}}_{e})}^{2} + {(y_{d} - {\bar{y}}_{e})}^{2}}

(19)

The precision of the location estimate can be quantified based on the area of the 95% error ellipse. Let $Σ$ denote the covariance matrix of vector $x_{e}$ and $λ_{1}$ and $λ_{2}$ its two eigenvalues. Then, the 95% confidence ellipse is defined as $P {{(x_{e}^{'} - μ_{e})}^{T} Σ^{- 1} (x_{e}^{'} - μ_{e}) \leq χ_{2}^{2} (0.95)}$ , where $χ_{2}^{2} (0.95) = 5.99$ . The axes of the ellipse are computed as $a_{1} = \sqrt{λ_{1} χ_{2}^{2} (0.95)}$ and $a_{2} = \sqrt{λ_{2} χ_{2}^{2} (0.95)}$ . The precision of the estimate is then computed as

\Pr = π a_{1} a_{2}

(20)

Simply picking the highest value of the localization algorithm may lead to erroneous readings due to signal noise. To reduce this effect, the location estimate presented in Ref [33] is adopted. The grid points whose values are above the 1% quantile are selected as possible damage location. Then, a circle is constructed that encloses the selected points. The damage location is to be at the center of the constructed circle. Effectively, this location estimate finds the center of the region with the highest localization values. For subsequent impact events, the values of the imaging algorithm are subtracted to estimate the location of the latest event. In Figure 16, the estimated location for each impact event is indicated with a black circle.

In Figure 17, the mean estimated damage locations are indicated for each impact event along with the 95% error ellipses Additionally, the accuracy and the precision computed for the localization of each impact event is reported in Table 7. As expected, impact events FP3-1 and FP3-4, that demonstrate the lowest damage indexes (see also Figure 15), have the lowest localization precision. The lowest accuracy is obtained for impact FP3-3 where the localization algorithm indicates that the impact was located below the stringer foot. Furthermore, the peak value of the imaging methodology is compared for each of the cases. Higher peak values are less likely to be confused with external signal degradation mechanisms (e.g., noise). The peak value is defined as the maximum value of the imaging algorithm within the enclosing circle constructed for the estimation of the damage location of each impact event. The location of each peak is indicated with a black cross in Figure 16 while the peak values are reported in Table 7.

Figure 17.

Damage localization results for all impact events on FP3 with $Δ T = 10 \circ C$ . Dashed lines indicate the error ellipses for each compensation methodology.

Table 7.

Summary of the computed accuracy and precision for each impact event.

	Case 2			Case 3			Case 4
Impact ID	$Acc [mm]$	$\Pr [m m^{2}]$	$Peak$	$Acc [mm]$	$\Pr [m m^{2}]$	$Peak$	$Acc [mm]$	$\Pr [m m^{2}]$	$Peak$
FP3 – 1	38	5837	0.50	17	1286	0.64	12	1248	0.75
FP3 – 2	16	119	1.17	8	86	1.38	9	110	1.38
FP3 – 3	46	260	0.53	38	385	0.58	35	66	0.63
FP3 – 4	8	3679	0.56	10	1586	0.72	10	2461	0.73
FP3 – 5	18	872	1.05	15	595	1.34	13	372	1.40
FP3 – 6	19	1171	0.70	6	437	0.79	6	73	0.83

Compared to Case 2, Cases 3 and 4 and can lead to improved damage localization. Use of the monostringer data for the compensation of the flat panel signals (Case 3) leads to an average improvement of 29% for the accuracy and 34.74% for the precision. This improvement can be further increased when the correction factors at the flat panel scale are allowed to be updated (Case 4) up to 32.5% for the accuracy and 57.5% for the precision. Cases 3 and 4 increase the peak value near the impact location by 20.7% and 27.9%, respectively, improving the reliability of the localization methodology. The results demonstrate the possible improvement in the accuracy and the precision of the damage localization algorithm.

Conclusions

Common temperature compensation approaches require the collection of large datasets that might be infeasible for large and complex structures. To alleviate this restriction, an up-scaling framework was proposed that leverages observations made at lower scale to improve the temperature compensation at the higher ones. The structural complexity of each scale was considered through the introduction of additional correction factors and a Bayesian regression model was used to estimate their values at different inputs and propagate the information collected between the scales.

First, the performance of the correction factors was demonstrated for measurements taken from monostringer specimens. Although the bonding process was not the same across all monostringers, the estimated correction factors were able to improve the temperature compensation compared to using the coupon compensation factors directly. Then, the correction factors were taken to a higher scale and were applied to the flat panel measurements. The proposed method was able to achieve a 7.19% reduction in the residual signals. Furthermore, through the Bayesian updating of the correction factors, the reduction can be increased up to 12.49%. Similar results were obtained when the data from the Co-Cured monostringer are used for the compensation of the Co-Bonded flat panels.

Significant improvement is also reported regarding the detection and localization of damage. Since the accuracy of the compensation is increased, the uncertainty in the definition of the threshold is reduced. On average, during damage detection, the Mahalanobis distance for the damaged cases presented was increased by 9.18% and 15.71% for Cases 3 and 4, respectively. During damage localization, the amplitude and time of arrival of the damage scattered signal was captured more accurately, improving the accuracy and the precision of the localization estimate. Case 3 improved the accuracy by 29% and the precision by 34.74% while Case 4 offers improvement up to 32.5% for the accuracy and 57.5% for the precision. Moreover, when the proposed framework is implemented, the imaging algorithm led to higher amplitude estimates (on average 20.74% and 27.85% for Cases 3 and 4, respectively) at the damage locations, increasing the reliability of the damage localization.

Although the scenario adopted in this study assumes only three scales (coupon-monostringer-flat panel), the framework can be applied to multiple distinct scales till the final structure. The specimens used in this study had similar features and sensor networks arrangements. Multiple paths within the same framework could also be created to account for a variety of structural features (such as curvature and change of thickness.) that might appear in a complete aircraft. The effect of temperature variation can be studied for such cases using smaller specimens at lower levels and a library of dedicated compensation and correction factors can be created for each one. A comparison between the different temperature compensation methodologies that have been proposed in the literature would be a potential research area for future investigations. Such comparison would not only illustrate the performance of each method but also highlight the requirements, benefits, and limitations of each method if up-scaling is attempted.

Footnotes

Appendix 1

In this Appendix, the imaging algorithm adopted for the localization of a damage event is described briefly. Damage localization can provide information regarding the existence of damage near critical areas and help the maintenance crew to focus subsequent Non-Destructive Testing (NDT) activities at specific areas, reducing the manual labor required. Various damage detection algorithms have been proposed in the literature.⁴⁸ In this study, localization is carried out by combining two approaches, the delay-and-sum (DAS)⁴⁹ and the reconstruction algorithm for the probabilistic inspection of defects (RAPID).¹⁷ The former algorithm aims at associating the damage scattered signals with a specific location of the structure based on its time of arrival and the wave propagation velocity. The later, projects the damage index computed between a sensor pair on its path and the location of the damage can be inferred when multiple paths that cross the same location demonstrate high damage indexes.

Implementing the DAS algorithm, the following imaging field can be obtained as

(A1)

I_{DAS} (x, y) = \frac{1}{n_{pairs}} \sum_{1}^{n_{pairs}} g (t_{i} (x, y))

where $g (t_{i}) = | (s_{i}^{ref} (t) - s_{i} (t)) - i H (s_{i}^{ref} (t) - s_{i} (t)) |$ is the envelope of the damage scattered signal, $H (\cdot)$ denotes the Hilbert transform, $t_{i} (x, y) = d_{i} (x, y) / d_{i} (x, y) c_{i} c_{i}$ is the arrival time at $(x, y)$ , and $d_{i} (x, y)$ is the distance from the actuator of the $i^{th}$ pair to $(x, y)$ and then to the sensor.

Next, an imaging field is obtained using the RAPID algorithm as

(A2a)

I_{RAPID} (x, y) = \sum_{1}^{n_{pairs}} (1 - D I_{i}) \frac{e - E_{i} (x, y)}{e - 1}

(A2b)

E_{i} (x, y) = {\begin{matrix} \begin{matrix} 1, & d_{i} / d_{i} d_{i} d_{i} (x, y) \geq e \end{matrix} \\ \begin{matrix} e, & d_{i} / d_{i} d_{i} d_{i} (x, y) < e \end{matrix} \end{matrix}

where $e = 0.975$ is a factor controlling the width of the ellipse that is defined between two transducers.

The RAPID algorithm can provide reliable localization results at the intersection of sensor pairs limiting the spatial resolution of the imaging technique. The DAS algorithm on the other hand is influenced by boundary reflections and wave attenuation at later time instants that can reduce its temporal resolution. To improve the imaging results, the two fields are combined at each pixel location using

(A3)

I (x, y) = I_{RAPID} (x, y) * I_{DAS} (x, y)

For more information on the localization approach, the interested reader is referred to Ref [9]and the references therein.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research leading to these results has gratefully received funding from the European JTICleanSky2 program under the Grant Agreement n° 314768 (SHERLOC). This project is coordinated by Imperial College London and lead by Leonardo S.p.A. as Topic Manager.

ORCID iDs

Ilias N. Giannakeas

Z. Sharif Khodaei

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