Pulsed terahertz spectroscopy combined with hybrid machine learning approaches for structural health monitoring of multilayer thermal barrier coatings

Dongdong Ye; Weize Wang; Changdong Yin; Zhou Xu; Haiting Zhou; Huanjie Fang; Yuanjun Li; Jibo Huang

doi:10.1364/OE.404042

1. Introduction

To achieve high thrust-weight ratio and combustion efﬁciency, ceramic thermal barrier coatings (TBCs) system is widely applied to assure the structure integrity of hot-section components of gas-turbine engines in the rigorous service environment. The surface layer of a typical atmospheric-plasma-sprayed (APS) TBCs system is generally the top brittle ceramic coating (TC), and is commonly a yttria-stabilized zirconia (YSZ) layer, owing to its superior mechanical properties, low-thermal conductivity, and high-phase stability; a vacuum-plasma-sprayed MCrAlY layer is usually chosen as the bond coat (BC) to improve the comprehensive mechanical and thermophysical abilities of the TBCs system [1,2]. There are many indicators that determine the service performance and lifetime of the TBCs, when the TC material has been confirmed, most of these indicators mainly depend on the structure properties of ceramic TC layer at design time, the thickness, porosity and roughness of TC enormously influence the comprehensive service performance of TBCs. Further, the TC thickness enormously influences the thermal insulation performance, the TC porosity has a huge effect on the fracture toughness, the TC roughness has strong bearing on the stress distribution and the interface bonding strength, to be more precise, these structure parameters are coupled with each other to determine the comprehensive service performance of TBCs. When the TBCs system is in service under high temperature and severe conditions, its original designed performance will continue to decline with service time, for instance, the internal pores of the TC layer get closed, the thickness of the TC layer is reduced, and the thermal insulation performance of the TC layer is also degraded, owing to the sintering effect in the long-term high-temperature service process. Therefore, it is necessary to conduct effective health monitoring on these structure parameters to prevent premature failure of TBCs system, not only that, it is also very vital for optimizing TBCs performance, inspecting service life, and developing novel TBCs [2–5].

In the last few decades, plentiful of nondestructive testing (NDT) methods had been conducted to evaluate the structural integrity of the TBCs system [6–9]. Compared to these above NDT methods, terahertz NDT technology has been considered as the most promising solution to evaluate the TBCs system in recent years, owing to its convenient, high-precision, nondestructive, noncontact, nonionizing, and rich feature information nature. Recent studies indicated that terahertz NDT technology had shown its superiority in the thickness measurement [10–12], porosity evaluation [13–15], and surface roughness estimation [16,17]. Nevertheless, these studies mainly used the variable-controlling approach to monitor the single characteristic quantity, while ignoring the variation of other structure parameters. The changes of these structure parameters in the actual service process are complicated, and it is not enough to recognize the variation of a certain parameter during monitoring, and the current development trend of TBCs shows that the single-layer YSZ ceramic top layer is no longer sufficient. To meet higher service requirements, it is necessary to introduce a multilayer ceramic top layer so that it can exert more excellent thermal insulation performance and ensure a longer service life [18,19]. Hence, the single-layer constitutive model established in the past cannot be applied to the multilayer ceramic top layer, and there is almost no literature report on the related research work of terahertz nondestructive testing that can simultaneously monitor the changes of multiple structure parameters. Therefore, it is necessary to re-conduct the theoretical analysis, re-establish the relevant constitutive models, and re-establish the extraction and measurement methods of each structure parameter feature to meet the non-destructive evaluation requirements of the multilayer ceramic coatings. This also poses the greater challenges and puts forward the higher requests to the terahertz NDT technology.

In our previous study, a multilayered YSZ TBCs with more excellent service performance and longer lifetime had been developed, and we took this case as an example to explore the multilayer TBCs structural health monitoring solution [20]. Theoretical analysis was conducted to solve equations simultaneously to extract structure parameters from terahertz time-domain spectrum and frequency-domain spectrum, the numerical simulation was conducted to obtain the terahertz time-domain signals by FDTD method, to get closer to the actual test signal, the experimental values of complex refractive indices of various YSZ TC layers which were estimated under the terahertz transmission system were imported into the FDTD simulation model, and the Gaussian white noise was used to add noise to the simulated time-domain signal, four wavelet noise processing methods were used to reduce the noise, and the principal component analysis (PCA) method was applied to reduce the data dimension after noise reduction. These signals processed by noise reduction and dimensionality reduction were adopted as the input feature so set up the extreme learning machine (ELM) modeling, and the Genetic algorithm (GA) was used to optimize the ELM model to increase its reliability and accuracy.

2. Theoretical analysis and numerical simulation

2.1 Time-frequency theoretical analysis

Previous studies had worked out the method that the thickness of the TC layer could be obtained by solving the simultaneous equations using the time-domain spectroscopy while the refractive index of the TC layer is unknown [11]. As shown in Fig. 1, here we can expand this work and extend its application to multilayer TC coatings. Terahertz waves are transmitted and reflected back and forth between the air, multiple TC layers, and BC. Multiple reflections can be obtained, supposing that the electric field of the incident terahertz waves has the frequency characteristics denoted as F₀, here we signed the first reflection on the subface of m_th TC layer as F_m and signed the second reflection on the surface of m_th TC layer as F_mm, hence, if the m=2 (double TC layer case), then the frequency characteristics of F₁, F₁₁, F₂, and F₂₂ could be estimated as follows:

(1)$${F_s} = {F_0} \cdot {r_{01}},$$

(2)$${F_1} = {F_0} \cdot {t_{01}} \cdot {t_{10}} \cdot {r_{12}} \cdot {h_1}^2,$$

(3)$${F_{11}} = {F_0} \cdot {t_{01}} \cdot {t_{10}} \cdot {r_{12}}^2 \cdot {r_{10}} \cdot {h_1}^4,$$

(4)$${F_2} = {F_0} \cdot {t_{01}} \cdot {t_{10}} \cdot {r_{23}} \cdot {t_{12}} \cdot {t_{21}} \cdot {h_1}^2 \cdot {h_2}^2,$$

(5)$${F_2}_2 = {F_0} \cdot {t_{01}} \cdot {t_{10}} \cdot {r_{23}}^2 \cdot {r_{21}} \cdot {t_{12}} \cdot {t_{21}} \cdot {h_1}^2 \cdot {h_2}^4.$$

Fig. 1. Multiple reflections of terahertz waves in the multiple TC layers (d: the thickness of each TC; m: the total number of TC layers).

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Solve the above equations simultaneously to reduce the h, the formulas were simplified as follows:

(6)$$\left|{\frac{{{F_s}}}{{{F_0}}}} \right|= \left|{\frac{{{n_1} - 1}}{{{n_1} + 1}}} \right|,$$

(7)$$\left|{\frac{{F_1^2}}{{{F_s} \cdot {F_{11}}}}} \right|= \left|{\frac{{4{n_1}}}{{{{({{n_1} - 1} )}^2}}}} \right|,$$

(8)$$\left|{\frac{{F_2^2}}{{{F_1} \cdot {F_{22}}}}} \right|= \left|{\frac{{4{n_1} \cdot {n_2}}}{{{{({{n_1} - {n_2}} )}^2}}}} \right|.$$

Here, the value of F could be obtained by fast Fourier transformation (FFT), n was the refractive index of each layer. Hence, the refractive index of each layer TC could be estimated, and then the time difference Δt between adjacent reflection peaks could be used to measure the thickness d of each layer as follows:

(9)$$d = \frac{{c\Delta t}}{{2n}}.$$

Here, c was the velocity of light in the vacuum.

Previous studies had proved that the refractive index was closely correlated with the porosity and they could convert to each other using effective medium theory (EMT) [15,21].

The above Eq. (8) can be extended iteratively to the case of m≥3, which can be rewritten as follows:

(10)$$\left|{\frac{{F_m^2}}{{{F_{m - 1}} \cdot {F_{mm}}}}} \right|= \left|{\frac{{4{n_{m - 1}} \cdot {n_m}}}{{{{({{n_{m - 1}} - {n_m}} )}^2}}}} \right|.$$

The above discussion focused on using time-domain spectroscopy to extract the structure parameter information, and the following was an analysis and discussion on the frequency-domain spectroscopy.

As shown in Fig. 2, for single TC layer, the reflection coefficient r of the each received reflection signal could be estimated as follows:

(11)$${r_1} = {r_{01}},$$

(12)$${\textrm{r}_2} = {\textrm{t}_{01}}{t_{10}}{r_{12}}{h_1}^2,$$

(13)$${\textrm{r}_3} = {\textrm{t}_{01}}{t_{10}}{r_{12}}^2{r_{10}}{h_1}^4,$$

(14)$${\textrm{r}_{^l}} = {\textrm{t}_{01}}{t_{10}}{r_{12}}^{\textrm{k} - 1}{r_{10}}^{\textrm{k} - 2}{h_1}^{2k - 2}(l = 4,5,6\ldots ,k = 4,5,6\ldots ).$$

Then the cumulative total reflection coefficient ${r_{\textrm{total }}}$ of the all these received reflection signal could be estimated as follows:

(15)$${r_{\textrm{total }}} = {r_1} + {r_2} + {r_3} + \cdots + {r_n} = \frac{{{r_{01}} - h_1^2}}{{1 - {r_{01}}h_1^2}}.$$

Here, $h = \exp ({ - \alpha d/2} )\exp ({i\delta } )$, $\alpha $ and $\delta $ were the absorption coefficient of TC and phase factor, respectively, and $\delta = 2\pi nd/{\lambda _a}$(${\lambda _a}$ was the wavelength in the vacuum). Hence, the reflectance ${R_{\textrm{total }}}$ could be estimated as follows:

(16)$${R_{\textrm{total }}} = {r_{\textrm{total }}} \times {r_{\textrm{total }}} = {\left( {\frac{{{r_{01}} - \textrm{exp}( - {\alpha_1}{d_1})\cos 2\delta }}{{1 - {r_{01}} \textrm{exp}( - {\alpha_1}{d_1})\cos 2\delta }}} \right)^2}.$$

Fig. 2. Reflection coefficients of the multiple TC layer

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From Eq. (16), it could be noted that the reflectance spectrum shows extreme peaks at $2\delta = 2p\pi$ and $2\delta = ({2\textrm{p } + \textrm{ }1} )\pi$ (p = 1, 2, 3…). The appearance of a series of extreme values gets the entire reflectance waveform to present the periodic oscillations, and the periodic interferential oscillations of reflectance spectrum merely depended on the optical thickness (nd). Therefore, the thickness and refractive index can be obtained by reproducing the positions of the extreme point and the period oscillations of the reflectance spectrum. This has been proven to be feasible in our previous study [15].

As shown in Fig. 2, for double TC layer, to solve the reflection problem of the multilayer TC layer, according to the principle of optical superposition, the double-layer TC could be simplified to a single-layer TC2’, the number of interfaces was gradually reduced, the recursive method was used to simplify the propagation of terahertz waves in the two interfaces 12 and 23 to the propagation in an interface 2'3, according to Eq. (21), the cumulative total reflection coefficient ${r_{total}}_{(m = 2)}$ could be estimated as follows:

(17)$${r_{total}}_{(m = 2)} = \frac{{{r_{01}} + {r^\prime }h_1^2}}{{1 - {r_{10}}{r^\prime }h_1^2}}\textrm{ = } = \frac{{({{r_{01}} + {r_{12}}h_1^2} )+ ({{r_{01}}{r_{12}} + h_1^2} ){r_{23}}h_2^2}}{{({1 + {r_{01}}{r_{12}}h_1^2} )+ ({{r_{12}} + {r_{01}}h_1^2} ){r_{23}}h_2^2}}.$$

Here, ${r^\prime }$ was the equivalent reflection coefficient of the interface 2'3. The positions of the extreme point and the period oscillations of the reflectance spectrum ${R_{total}}_{(m = 2)}$ depended on the optical thickness (${n_1}{d_1}$ and ${n_2}{d_2}$) of TC1 and TC2. For m≥3, by that analogy, the positions of the extreme point and the period oscillations of the reflectance spectrum ${R_{total}}_{(m\ge3)}$ depended on the optical thicknesses (${n_1}{d_1}$, ${n_2}{d_2}$ … ${n_m}{d_m}$) of TC1, TC2 …. TCm.

It should be noted that the effect of roughness was not considered in the theoretical analysis of either time-domain spectroscopy or frequency-domain spectroscopy. In fact, when the terahertz wave was reflected on the rough surface, its effective reflectance could be estimated by Kirchhoff’s law, which was also a method used in the Refs. [16,17] to determine the surface roughness of TBCs, the actual effective reflectance ${R_{\textrm{rough }}}$ could be estimated as follows:

(18)$$\frac{{{R_{\textrm{rough }}}}}{{{R_{\textrm{smooth }}}}} = \textrm{exp} \left( { - {{\left( {\frac{{4\pi \sigma }}{\lambda }} \right)}^2}} \right).$$

Here ${R_{\textrm{smooth }}}$ was the specular reflectance, $\sigma $ was the roughness, and λ was the terahertz wavelength.

If the roughness factor was added for consideration, the above analysis need to consider the multi-interface roughness to modify the above equations, so whether the time domain spectrum method or the frequency domain reflectance spectrum method was chosen, after considering the roughness, the former method cannot be directly performed by extracting the time domain reflection peak to obtain multiple structure parameters, since the thickness of the actual each TC layer was very thin and the adjacent reflection peaks were superimposed together, hence it was almost impossible to extract each own exclusive reflection peak, even though the most advanced signal processing method was used, it may only be used to pick the time point position of the peak to obtain the adjacent time difference Δt, but it cannot obtain other parameters (such as the refractive index and roughness). Although the latter method can theoretically obtain the corresponding extreme points, oscillation period and reflectance spectrum intensity, which could be used to return the structure parameters through the waveform inversion method. Nevertheless, there were more TC layers and more structure parameters in multilayer TBCs, more than one solution may be obtained, not only that, the method was computationally expensive, overly dependent on experience, and low reliability, hence, it was not suitable for actual multilayer TBCs structure health monitoring.

2.2 FDTD numerical simulation

The FDTD algorithm was applied to discretize the Maxwell curl equation in the time and space domains, and took the central difference approximation to the first-order partial derivatives of time and space to find the approximate solution of the Maxwell curl equation. In the three-dimensional Cartesian coordinate system, the time-domain recursion method of the E_x component in Maxwell’s equation was described as follows:

(19)$$\begin{aligned}&{E_x^{n + 1}\left( {i + \frac{1}{2},j,k} \right) = \frac{{1 - \frac{{\Delta t \cdot \sigma }}{{2\varepsilon }}}}{{1 + \frac{{\Delta t \cdot \sigma }}{{2\varepsilon }}}}E_x^n\left( {i + \frac{1}{2},j,k} \right) + \frac{{\Delta t}}{{\delta \varepsilon \left( {1 + \frac{{\Delta t \cdot \sigma }}{{2\varepsilon }}} \right)}}\left[ {H_z^{n + \frac{1}{2}}\left( {i + \frac{1}{2},j + \frac{1}{2},k} \right)} \right.}\\ &{\left. { - H_y^{n + \frac{1}{2}}\left( {i + \frac{1}{2},j - \frac{1}{2},k} \right) - H_y^{n + \frac{1}{2}}\left( {i + \frac{1}{2},j,k + \frac{1}{2}} \right) + H_z^{n + \frac{1}{2}}\left( {i + \frac{1}{2},j,k - \frac{1}{2}} \right)} \right]} \end{aligned}.$$

Here, $\Delta t$ was the time step, $\varepsilon $ was the dielectric constant, and $\sigma $ was the electrical conductivity, $\delta $ was the space step, $\textrm{(i, j, k)}$ was the node of the Yee unit, n was the time step of the calculation. The equations of other components of the electromagnetic field were all derived in the similar way, so it is not necessary to repeat these equations. For a three-dimensional cube Yee cell, the side length of the cube is the space step. For the details on maintaining numerical stability and numerical dispersion conditions, please refer to the Ref. [22].

In this work, the FDTD Solutions software (Lumerical Solutions Inc., Vancouver, Canada) was used to model and simulate the propagation process of terahertz waves in multilayer TBCs, the incident terahertz Gaussian waves with the broadband radiation were set perpendicular to the TBCs surface.

The effective dominant frequency of terahertz waves was set in the range of 0.3-1.5 THz. Periodic boundary conditions were set in the X and Y directions, and perfectly matched layer (PML) boundary condition was set in the terahertz incidence Z direction, much more details about FDTD simulation of terahertz wave interacting with multilayer coatings could be discovered in Refs. [22,23]. By investigating the structure parameters of the triple layer TBCs in Ref. [20], the parameter settings for the thickness, porosity and roughness of each layer during the FDTD simulation were shown in the Table 1. It should be explained that there was a total of four surface interfaces among the three TC layers. To simplify the FDTD simulation calculation, we set the interface roughness of the middle two layers (YSZ1-TC1 / YSZ2-TC2, YSZ2-TC2 / YSZ3-TC3) to the same roughness value and named it the internal roughness. The roughness of the other two layers were named the surface roughness (the surface roughness of TC1) and interface roughness (the bottom roughness of TC3), respectively. Herein, there were four selectable variables for each structure parameter of each TC, which meant that there was a total number of 64 permutations for each structure parameter (4 × 4 × 4 = 64), the three 64 sets of simulation parameters were combined and imported into the FDTD simulation models, so there were a total of 64 models to be calculated.

Table 1. Parameter settings for the thickness, porosity, and roughness of each YSZ layer

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In addition to the thickness and roughness, the porosity could not be directly imported into the FDTD model and need to be converted into the corresponding complex refractive index ($\tilde{n}(\omega ) = n + ik$, n was the refractive index and k was the extinction coefficient) into the model for calculation, the YSZ ceramic pellets with various porosities were prepared by adjusting the YSZ powder particle size, spraying power and spraying distance, the details of the preparation process could be found in our previous research papers [10,14,15], in this work, it was not our focus. The complex refractive index was estimated using a TAS7400TS transmission terahertz time-domain spectroscopy system (Advantest Corporation, Japan. The wavelength was 1560nm, the pulse width was 300 fs, and the frequency resolution was 1.9 GHz), the spot diameter of terahertz waves was 2 mm, each complex refractive index data was obtained by 256 scans that were averaged to ensure the data reliability, three different disjoint areas near the pellet center were tested for each sample.

From the transmission reference and sample signals, the frequency-domain function E_reference (ω) and E_sample (ω) were obtained by taking the fast fourier transform (FFT). Therefore, the complex transmittance transfer function $\hat{H}(\omega )$ was expressed as follows [24,25]:

(20)$$\hat{H}(\omega ) = \frac{{{E_{reference}}(\omega )}}{{{E_{sample}}(\omega )}} = \frac{{4\tilde{n}(\omega )}}{{{{[(\tilde{n}(\omega ) + 1)]}^2}}}\exp \{{ - i} \left. {\frac{{[(\tilde{n}(\omega ) - 1)]d\omega }}{c}} \right\}$$

where ω and c were the frequency of terahertz waves and the speed of light in vacuum, respectively. Hence, the formulas for calculating the refractive index n and extinction coefficient k of sample sere deduced as follows:

(21)$$n = \frac{{c\varphi (\omega )}}{{\omega d}} + 1$$

(22)$$\textrm{k} = \frac{c}{{\omega d}}\ln \left( {\frac{{|{{E_{reference}}(\omega )} |}}{{|{{E_{sample}}(\omega )} |}}\cdot \frac{{4n(\omega )}}{{{{[{(n(\omega ) + 1)} ]}^2}}}} \right)$$

where $\varphi (\omega )$ was the phase difference between reference and sample signals.

To mimic the actual terahertz signal in structural health monitoring of multilayer thermal barrier coatings as much as possible, the Gaussian white noise with a SNR of 16 dB was added into the simulated time-domain signal. Despite the ideal simulation signal with Gaussian white noise could not accurately represent the noise oscillation interference in actual test, it was a commonly accepted method employed in the theoretical study of the terahertz NDT research [26,27].

3. Signal processing and machine learning modeling

In this work, four wavelet denoising methods were used to reduce noise: dmey global default threshold for denoising, sym3 wavelet heuristic SURE threshold denoising, haar wavelet soft SURE threshold denoising, db3 wavelet fixed threshold denoising (their decomposition levels were set to 7) [28,29]. The reference signal without TC layer was still taken as an example, the residual analysis was performed using the denoised signal and the original unnoisy signal, and the variation of the residual was used to determine which type of noise reduction method was the appropriate noise reduction method, that was, the noise reduction method adopted in this study.

The ELM is a new type of feed-forward neural network with n training samples $\{{({{x_s},{t_s}} )} \}_{s = 1}^n$(${x_s}$ is the input vector and ${t_s}$ is the output vector). An ELM regression model containing m hidden layer neuron functions $f({\cdot} )$ can be expressed as follows [30,31]:

(23)$$\left\{ {\begin{array}{l} {\sum\limits_{i = 1}^m {{\omega^\textrm{T}}} f({{w_{\textrm{in}}}{x_1} + {b_i}} )= {t_1}}\\ {\sum\limits_{i = 1}^m {{\omega^\textrm{T}}} f({{w_{\textrm{in}}}{x_2} + {b_i}} )= {t_2}}\\ \vdots \\ {\sum\limits_{i = 1}^m {{\omega^\textrm{T}}} f({{w_{\textrm{in}}}{x_s} + {b_i}} )= {t_s}} \end{array}} \right.$$

where s is the number of training samples; ${w_{\textrm{in}}}$ is the input weight of input node and hidden layer node; $\omega $ is the output weight value connecting the hidden layer and the output layer; ${b_i}$ is the deviation of the i_th neuron, that is, the hidden layer threshold. Equation (23) can be converted to matrix form, which can be expressed as follows:

(24)$$\textbf{H}\textrm{W = }\textbf{T}$$

(25)$$\textbf{H} = {\left[ {\begin{array}{ccc} {f({{w_{\textrm{in}}}{x_1} + {b_1}} )}& \cdots &{f({{w_{\textrm{in}}}{x_1} + {b_l}} )}\\ \vdots &{}& \vdots \\ {f({{w_{\textrm{in}}}{x_n} + {b_1}} )}& \cdots &{f({{w_{\textrm{in}}}{x_n} + {b_l}} )} \end{array}} \right]_{N \times l}}$$

(26)$$W = {[{{\omega_1},{\omega_2} \cdots {\omega_L}} ]^\textrm{T}}.$$

Here, H is the output matrix of the hidden layer of the neural network; W is the output weight; T is the output vector, $T = [{{t_1},{t_2}} { { \cdots ,{t_k}} ]^\textrm{T}}$; Since it is much larger than m in most cases. Then the output weight value can be expressed as follows:

(27)$$w = {({{\textbf{H}^\textrm{T}}\textbf{H}} )^{ - 1}}{\textbf{H}^\textrm{T}}\textbf{T}.$$

Therefore, the ELM time series prediction model finally obtained after training can be expressed as follows:

(28)$$t = \sum\limits_{i = 1}^m {{w_i}} f({{w_{1n}}x + {b_i}} )$$

where x is the input of the prediction model and t is the output of the prediction model.

Compared with the classical back-propagation (BP) neural network model, ELM has the faster learning speed, stronger generalization performance, and is not easy to fall into local extremes. However, there are still some problems, such as the random selection of input weights and hidden layer bias in the ELM algorithm, which leads to implicit layer neurons have almost no adjustment ability, this puts forward higher requirements for the optimization of network parameters. Genetic algorithm (GA) is a parallel random search optimization method formed by simulating the genetic mechanism of nature and species evolution. It encodes the parameters that need to be optimized to form a tandem population, and then selects the best individual through selection, crossover and mutation according to the fitness function. GA is used to optimize the input weight and hidden layer bias of the ELM neural network to obtain the optimal network parameters to establish the GA-ELM neural network to improve the accuracy and stability of the model output [32,33]. The GA-ELM algorithm integrates the GA global search optimal ability and the strong learning ability of ELM. In this algorithm, the input weights and hidden layer node offsets of ELM training data are mapped to genes on each chromosome in the GA population, and the chromosome fitness of GA corresponds to the training error of ELM. The problem of obtaining the optimal input weight and bias is transformed into the problems of calculating chromosome fitness and selecting the optimal chromosome. Figure 3 shows the GA-ELM algorithm flow, which mainly includes the ELM network determination, the genetic algorithm optimization, and the ELM network training and prediction. The specific implementation steps are as follows [34,35]:

1) Population initialization. Determine the topological structure of the ELM neural network, that is, the number of neurons in the input layer, hidden layer, and output layer; set the maximum evolution algebra G; randomly generate the input weights and hidden layer bias of the ELM neural network, and perform binary encoding generates the initial population; the length of the individual is composed of the hidden layer input weight matrix and bias vector, that is, D=(n+1)L, L is the number of hidden layer nodes, n is the number of neurons in the input layer, that is, the input vector dimension.
2) Evaluation of individual fitness. For any individual in each generation of the population, the ELM algorithm is used to calculate the output weight matrix, and the root mean square error (RMSE) between the expected output and the actual output is obtained as the GA objective function. The smaller the value of the objective function, the more accurate the model.
3) Population evolution. According to the fitness of the individual, the roulette method is used to select the chromosomes in each generation of the population, and the probability-based crossover and mutation operations are used to optimize the selected individuals to generate a new population until the constraints are met. If the maximum number of iterations is reached, or the average target value and minimum target value of adjacent populations change very little, the evolution is terminated, and the final population is obtained.
4) ELM network training and prediction. Decode the final population after iterative optimization, get the optimized input weights and biases, and assign them to ELM model, use training samples to train the ELM network, use the least squares method to calculate the output layer weights, and finally bring the test samples into ELM model to perform predictive output.

Fig. 3. Calculation flow of GA-ELM algorithm

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In this study, for the standard ELM model, the sigmoid activation function was selected, and the number of hidden layer neurons was set to 40. For the convenience of comparison, the GA-ELM model chose the same activation function and the number of hidden layer nodes. The GA parameters were set as follows: the population size was 20, the maximum genetic generation number was 500, the crossover probability was 0.7, the mutation probability was 0.01, and the generation gap was 0.95. The 54 random samples were employed as the training sets to set up PCA-GA-ELM models, and the remaining 10 samples were used as the prediction sets. The reliability and accuracy of the suggested models were assessed using two evaluation indicators objectively, including the RMSE and the squared correlation coefficient (R). Their definitions were as follows:

(29)$$RMSE = \sqrt {\sum\limits_{i = 1}^n {{{({{Y_i} - {{\hat{Y}}_i}} )}^2}} /n} $$

(30)$$ R=\frac{\sum_{i=1}^{n}\left(\hat{Y}_{i}-\overline{\hat{Y}}\right)\left(Y_{i}-\bar{Y}\right)}{\sqrt{\sum_{i=1}^{n}\left(\hat{Y}_{i}-\overline{\hat{Y}}\right)^{2}} \sqrt{\sum_{i=1}^{n}\left(Y_{i}-\bar{Y}\right)^{2}}} $$

where n was the sample size, ${Y_i}$ was the real value of oxide scale thickness, ${\hat{Y}_i}$ was the predicted value of oxide scale thickness estimated by machine learning model.

The principal component analysis (PCA) algorithm is a data analysis method and its starting point is to calculate a set of new features arranged in descending order of importance from a set of features. PCA method is to project from a high-dimensional data space to a low-dimensional data space along the direction of maximum covariance, each principal component (PC) obtained is a linear sum of the original variables. The PCs obtained by the PCA are independent of each other. Reduce the dimension of data while ensuring that the loss of original data information is as small as possible. The PCA method can not only reduce the dimension of high-dimensional data, but also eliminate the redundant information in high-dimensional data. In this study, the PCA function was implemented through MATLAB programming. The 64 sets of noise-added terahertz time-domain and reflectance data were denoised by the wavelet denoising method. After PCA dimensionality reduction, the eigenvalues, contribution rate, and score of each principal component were obtained. These PCs were used to replace the original terahertz time-domain and reflectance spectral data as input features during modeling [30,36,37]. To further verify the robustness and accuracy of the hybrid PCA-GA-ELM model, 8-fold cross-validation (CV) was applied to divide the whole dataset into 8 folds. Combined with the characteristics of the sample number obtained in this study, it is suggested that the computation time and bias were less when 8-fold cross-validation was chosen [38,39]. The training of the model was performed on the 7 folds, and the validating was performed on the remaining 1 fold. The training and validating process was repeated several times by using different subsets as the validating sets. The overall prediction accuracy of the model is obtained by averaging the accuracy of each fold in the 8 training-validating rounds [40].

4. Results and discussion

According to the time-frequency theoretical analysis in section 2.1, the time-domain spectrum and the frequency-domain reflectance spectrum embodied the structure parameter messages (thickness, porosity and roughness) of the different multilayer TC, whereas these structure parameter messages were unlikely obtained by analyzing the time-frequency spectrums, owing to the redundant data embedded in these spectrums, these structure features were buried in these redundant data and became difficult to be extracted.

As shown in Fig. 4, the complex refractive indices of 16 YSZ ceramic pellet samples with the porosity ranged from 6.36% to 22.62% were estimated and employed as the imported optical properties of the three TC layers during FDTD modeling.

Fig. 4. Complex refractive indices of YSZ ceramic pellet samples at different porosities: (a) refractive index; (b) extinction coefficient.

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As shown in Fig. 5, the ideal simulation reference signal was processed by adding Gaussian white noise into it to obtain a noise superimposed signal, and it was obvious that the noise superimposed signal in Fig. 5(c) was much closer to the actual test signal curve. Such a signal containing high-frequency noise further increased the difficulty of structure parameter message extraction in actual nondestructive testing. Furthermore, the formula solving method derived from the above theoretical analysis was almost unusable.

Fig. 5. The process of adding Gaussian white noise to the FDTD simulation signal (took the reference signal as an example) : (a) ideal simulation reference signal; (b) Gaussian noise signal; (c) noise superimposed signal.

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As shown in Fig. 6, the residual distributions (a) ∼ (d) were applied to identify which denoising method was better, it could be seen that the denoising effect of the sym3 wavelet was the worst, and its residual value was the largest. The effects of the other three wavelet denoising methods were excellent, of which haar wavelet denoising method was the best, and the residual obtained by haar wavelet denoising method was the smallest and the residual distribution law was more consistent with the original set noise signal oscillation law, hence, the haar wavelet denoising method was adopted to perform noise reduction on 64 sets of noise-added data.

Fig. 6. The residual distributions (a) ∼ (d) (took the reference signal as an example): (a) dmey wavelet; (b) sym3 wavelet; (c) haar wavelet; (d) sym3 wavelet.

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As shown in Fig. 7, the terahertz time-domain spectrums and effective reflectance spectrums of the samples numbered 3 and 4 were obtained by FDTD calculation after haar wavelet denoising, and the effective reflectance spectrum ${R_{\textrm{effective}}}$ was estimated as follows:

(31)$${R_{\textrm{effective}}} = {\left[ {\frac{{|{\textrm{FFT}({{E_{\textrm{sam }}}(t)} )} |}}{{|{\textrm{FFT}({{E_{\textrm{ref}}}(t)} )} |}}} \right]^2}.$$

Here, ${E_{\textrm{sam }}}(t)$ and ${E_{\textrm{ref}}}(t)$ were the terahertz time domain signals with TBCs sample and reference sample, respectively.

Fig. 7. Terahertz spectrums of samples numbered 3 and 4 : (a) time domain spectrum; (b) effective reflectance spectrum.

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As shown in Fig. 7, taking the third layer TC as an example, small changes in the third layer TC structure parameters would cause slight changes in the time-frequency waveforms, and the changes were more obvious in the reflectance spectrum, but this still cannot be directly used to extract the specific quantitative information of the structure parameters through the slight changes in the time and frequency spectrums and machine learning modeling methods could help us to solve this problem.

The terahertz time and frequency data dimensions of 64 samples obtained by the FDTD simulation were 64 × 4035 and 64 × 202 (∼ 2 THz), respectively. If these data were directly used for modeling, the calculation speed would be reduced and the prediction accuracy of regression would also be reduced. Hence, PCA algorithm was used to reduce the dimensions of terahertz time-frequency data. As shown in Fig. 8, to make the cumulative contribution rate reached 95%, it was found that both the cumulative contribution rate of the first 25 components and the first 14 principal components of time-domain data and frequency-domain data were over 95%. Finally, the data dimensions were reduced from 64 × 4035 and 64 × 202 to 64 × 25 and 64 × 14, respectively, these dimension-reduced data were used as the input to ELM and GA-ELM models for the multilayer TBCs structural health monitoring.

Fig. 8. The contribution rate of each principal component and their cumulative contribution rate of the principal components: (a) time domain data; (b) frequency domain data.

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As shown in Fig. 9, the time-domain data after dimensionality reduction was used for model training, these fitness evolution curves indicated that when these trained models evolved to the 231, 289 and 51 generations, respectively, the training errors of the three structure parameters reached to their smallest values and met the trained requirements.

Fig. 9. Fitness evolution curve (trained by time-domain data): (a) thickness; (b) porosity; (c) roughness.

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Fig. 10. The prediction comparison results of the three structure parameters of the PCA-ELM model and PCA-GA-ELM model (predicted by time domain data): (a) YSZ1 thickness; (b) YSZ2 thickness; (c) YSZ3 thickness; (d) YSZ1 porosity; (e) YSZ2 porosity; (f) YSZ3 porosity; (g) surface roughness; (h) internal roughness; (i) interface roughness.

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Combined with the prediction comparison results of three structure parameters of the PCA-ELM model and PCA-GA-ELM model (predicted by time domain data) showed in Fig. 10(a)–(i), it could be concluded that the prediction performance of the PCA-ELM model was very unstable and poor; while the prediction performance of the PCA-GA-ELM was better, their R values of every structure parameters exceeded 0.93, and their RMSE values of every structure parameters were maintained at the low level.

Similar to the above time domain data, as shown in Fig. 11, the reflectance data after dimensionality reduction was used for model training, these fitness evolution curves indicated that when these trained models evolved to the 465, 315 and 412 generations, respectively, the training errors of the three structure parameters reached to their smallest values and met the trained requirements.

Fig. 11. Fitness evolution curve (trained by frequency domain data): (a) thickness; (b) porosity; (c) roughness.

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As shown in Table 2, the prediction comparison results of three structure parameters of the PCA-ELM model and PCA-GA-ELM model (predicted by reflectance data) indicated that the prediction performance of the PCA-ELM model was still very unstable, and the prediction performance was also unstable and poor. Similar to the above PCA-GA-ELM prediction results (predicted by time domain data), the prediction performance of the PCA-GA-ELM were also good, the values of R exceeded 0.92, and the values of RMSE were also maintained at the low level.

Table 2. The prediction comparison results of the three structure parameters of the PCA-ELM model and PCA-GA-ELM model (predicted by frequency domain data)

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To further verify the prediction accuracy and robustness of the hybrid models, as shown in Table 3 and Table 4, similar to the above prediction results without k-fold CV, the prediction comparison results of three structure parameters of the PCA-ELM model and PCA-GA-ELM model obtained by 8-fold CV (predicted by both the time domain data and frequency domain data) indicated that the prediction performance of the PCA-GA-ELM were still good, the values of R exceeded 0.90, and the values of RMSE were also maintained at the low level. This proved that the of our proposed hybrid PCA-GA-ELM model was possessed of good robustness and high reliability.

Table 3. The prediction comparison results of the three structure parameters of the PCA-ELM model and PCA-GA-ELM model obtained by 8-fold CV (predicted by time domain data)

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Table 4. The prediction comparison results of the three structure parameters of the PCA-ELM model and PCA-GA-ELM model obtained by 8-fold CV (predicted by frequency domain data)

View Table | View all tables in this article

Unlike the prediction results of YSZ1 obtained by PCA-ELM model (predicted by time domain data), the values of R (predicted by reflectance data) exceeded 0.94, this proved that the reflectance spectrum was more sensitive to the three structure parameters of YSZ1 than the time domain spectrum, it showed the superiority of reflectance spectrum in structure health monitoring to some extent, but for YSZ2 and YSZ3, both the regression prediction results obtained by the time domain data and the frequency domain data combined with the PCA-ELM were still poor.

The optimization algorithms PCA and GA got the ELM model calculation speed improved and made the ELM model robustness and prediction precision have a strong guarantee, and the prediction performance of the time domain data and reflectance data were similar to each other. Nevertheless, compared to the data dimensions of both the time-domain data and the reflectance data, the data volume of the latter was 44% lower than the former, therefore, when using reflectance data for modeling, it would help to further improve the prediction speed. Finally, a double increase in prediction robustness, accuracy and speed could be achieved.

5. Conclusions

In this work, the terahertz nondestructive technology was proposed to monitor the structural health of multilayer thermal barrier coatings. Firstly, the time-frequency theoretical analysis was conducted to reveal the relationship among the TC structure parameters, the time domain waveform and the reflectance waveform. The theoretical analysis results indicated that even though both the two waveforms embodied the structure parameter messages of multilayer TBCs, but they could not be directly adopted to extract the structure parameter messages by solving simultaneous equations or the waveform inversion, owing to the multilayer complex TC structure and very thin TC layer. Secondly, the FDTD numerical simulation method was adopted to obtain the terahertz time-domain signals, 64 samples which were imported with three kinds of 64 sets of simulation structure parameters had been calculated to obtain the time domain signal, to mimic the testing signal, the Gaussian white noise with a SNR of 16 dB was added to the FDTD simulated terahertz time-domain signal. Four wavelet denoising methods were conducted to reduce the added Gaussian white noise, by comparing the residual difference between the filtered signal and the original unnoisy signal, haar wavelet denoising method achieved the best noise reduction performance. To further eliminate redundant information and improve calculation efficiency during modeling, the PCA method was successfully to reduce the dimensions of time-domain data and reflectance data from 64 × 4035 and 64 × 202 to 64 × 25 and 64 × 14, since their cumulative contribution rates of the first 25 and 14 principal components reached by over 95%. Thirdly, the PCA-ELM and PCA-GA-ELM models were set up and trained to monitor the structure health using time-domain data and reflectance data after the haar wavelet noise reduction processing. The prediction results of both the two machine learning models were evaluated by two indicators RMSE and R, 8-fold cross-validation was adopted to further verify the robustness and accuracy of the proposed hybrid model. The evaluation and comparison results indicated that the proposed hybrid PCA-GA-ELM model showed good performance in terms of structure health monitoring in both the prediction performance and calculation speed, reflectance data was more suitable for modeling. Finally, we believe our proposed novel hybrid machine learning method combined with the terahertz nondestructive technology will play a more important role in monitoring the structural integrity of TBCs.

Funding

Science and Technology Commission of Shanghai Municipality (16DZ2260604); National Natural Science Foundation of China (51775189).

Acknowledgments

We thank the support from the Shanghai Institute of Applied Physics, Chinese Academy of Sciences. We would like to thank professor Fangming Shao for the modeling guidance on our work.

Disclosures

The authors declare no conﬂicts of interest.

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TC layer	Optional thickness (µm)	Optional porosity	Optional roughness (Ra / µm)
YSZ1-TC1	30, 40, 50, 60	6.36%, 7.32%, 9.43%, 11.56%	2, 6, 10, 14
YSZ2-TC2	110, 120, 130, 140	19.54%, 20.59%, 21.65%, 22.62%	2, 6, 10, 14
YSZ3-TC3	70, 80, 90, 100	12.7%, 15.31%, 16.43%, 18.58%	2, 6, 10, 14

Performance indicator	d1	d2	d3	η1	η2	η3	σ1	σ2	σ3
RMSE-PCA-ELM	3.2873	12.8851	13.1086	0.0057	0.0121	0.0232	1.5586	5.0269	5.1322
RMSE-PCA-GA-ELM	2.6896	4.0580	6.2285	0.0041	0.0040	0.0097	0.7826	1.9555	1.6278
R-PCA-ELM	0.9512	0.4749	0.5628	0.9424	0.5520	0.4840	0.9447	0.4978	0.4125
R-PCA-GA-ELM	0.9665	0.9502	0.9279	0.9728	0.9199	0.9364	0.9874	0.9163	0.9433

Performance indicator	η1	η2	η3	d1	d2	d3	σ1	σ2	σ3
RMSE-PCA-ELM	0.8820	1.6707	1.7439	4.5217	5.2816	6.4884	0.0032	0.0045	0.0100
RMSE-PCA-GA-ELM	0.5025	0.7516	0.4358	0.9974	1.5139	1.1849	0.0024	0.0014	0.0027
R-PCA-ELM	0.7428	0.5129	0.6492	0.6948	0.4758	0.5813	0.7839	0.6903	0.2956
R-PCA-GA-ELM	0.9328	0.9527	0.9343	0.9199	0.9135	0.9374	0.9626	0.9085	0.9302

Performance indicator	η1	η2	η3	d1	d2	d3	σ1	σ2	σ3
RMSE-PCA-ELM	0.4869	3.0466	2.8365	1.2143	3.5212	5.1087	0.0021	0.0037	0.0042
RMSE-PCA-GA-ELM	0.5725	0.6508	0.3539	0.8091	1.2946	1.7941	2.98E-4	6.02E-4	0.0020
R-PCA-ELM	0.9441	0.4115	0.4924	0.9574	0.3256	0.1816	0.9683	0.0910	0.4021
R-PCA-GA-ELM	0.9768	0.9101	0.9071	0.9540	0.9103	0.9178	0.9990	0.9412	0.9518

TC layer	Optional thickness (µm)	Optional porosity	Optional roughness (Ra / µm)
YSZ1-TC1	30, 40, 50, 60	6.36%, 7.32%, 9.43%, 11.56%	2, 6, 10, 14
YSZ2-TC2	110, 120, 130, 140	19.54%, 20.59%, 21.65%, 22.62%	2, 6, 10, 14
YSZ3-TC3	70, 80, 90, 100	12.7%, 15.31%, 16.43%, 18.58%	2, 6, 10, 14

Pulsed terahertz spectroscopy combined with hybrid machine learning approaches for structural health monitoring of multilayer thermal barrier coatings

Abstract

1. Introduction

2. Theoretical analysis and numerical simulation

2.1 Time-frequency theoretical analysis

2.2 FDTD numerical simulation

3. Signal processing and machine learning modeling

4. Results and discussion

5. Conclusions

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (11)

Tables (4)

Equations (31)

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