Damage Detection of Refractory Based on Principle Component Analysis and Gaussian Mixture Model

1The State Key Laboratory of Refractories and Metallurgy, Wuhan University of Science and Technology, Wuhan 430081, China 2Key Laboratory of Metallurgical Equipment and Control Technology, Wuhan University of Science and Technology, Wuhan 430081, China 3Ministry of Education, Hubei Key Laboratory of Mechanical Transmission and Manufacturing Engineering, Wuhan University of Science and Technology, Wuhan 430081, China 4National Demonstration Center for Experimental Machinery Education, Wuhan University of Science and Technology, Wuhan 430081, China 5Smart Materials and Structures Laboratory, Department of Mechanical Engineering, University of Houston, Houston, TX 77204, USA


Introduction
Structural health monitoring (SHM) has made significant advances in the past decades [1][2][3][4][5][6][7][8].Monitoring of refractories, which are widely used in furnace, iron, and steel industries due to their ability to gain strength rapidly and to withstand aggressive environments and high temperature [9], receives increasing attention [10][11][12][13].Temperature variations can lead to either interfacial separation between aggregates and matrix or microcracks, both depending on the range of coefficient of thermal expansion (CTE) mismatch between phases.Such effects modify all the thermomechanical properties of the material, especially Young's modulus (E) [10,11,14].The AE technique has been developed over the last two decades as a nondestructive evaluation technique and as a useful tool for material research [15][16][17].It is an efficient method to monitor, in real time, damage growth in both structural components and laboratory specimens.This technique was often used to detect Young's modulus because it was correlated to AE activity variations considering the specific types of damage induced by CTE mismatch [12,13].The acoustic emission technique and the ultrasonic pulse echography technique, both carried out at high temperature, were applied as nondestructive characterization methods to monitor the damage extension within the materials submitted to thermal stress and to follow the evolution of the associated elastic properties [18,19].With this as a basis, the study could provide an important reference for thermal stress analysis under the AE data processing method.However, the AE signals generated by the complex structure of the refractory 2 Complexity are extremely complex even at normal temperature, which makes it difficult for the damage classification [20].For this purpose, the AE signal parameters of the delay distribution, rise time, energy, and peak amplitude were selected to distinguish the effective features for different failure mechanism so that the two failure modes of fiber breakage and delamination can be distinguished [21,22].The related parameters can be modeled by a generative model, in particular a Gaussian mixture model (GMM) in the field of dimension processing [23,24].The global feature descriptor was formed by stacking the parameters of the adapted GMM (i.e., means, covariance, and weight) in a so-called supervector [25,26].Also, some scientists paid more attention to the parameter of the signal energy moment compared to the peak amplitude distribution in the study of the glass fiber composite materials and chose it to distinguish the fiber breakage and debonding crack.Moreover, the amplitude, ring count, and felicity ratio were found more suitable in the damage study of the B-Al composite [27].However, much effort was put on the characterization of the overall parameters rather than on the data analysis of the damage mechanism.
Optionally, the dimensionality of the feature vectors can be reduced by a principal component analysis (PCA) [28].The PCA was used to generate a new set of noncorrelated features to remove interference and to avoid using low variance variables (that was almost single-valued variables).Moreover, these new features were selected according to their discriminative capability.Subsequently, feature space modeling and classification were addressed by means of probabilistic selforganizing maps (SOM), a fuzzy version of classical SOM that allowed measuring the activation probability of each unit [29,30].Nevertheless, detecting not only an event but also the type was not a straightforward task, and previous approaches had not been able to obtain high per attack detection accuracy values.Scientists showed that the resulting GMM supervector encoding yielded an excellent representation for fuzzy parameters [31,32].This method was an outstanding technique for handling the description of multimodal data, making it robust with high computational efficiency [26].Additionally, scientists employed support vector machines (SVM) to build individual classifiers per sample cluster [33,34].Such a SVM was a linear classifier trained by only one single positive sample and multiple negative samples; it was denoted as Exemplar-SVM.Therefore, secondly, using the features extracted by AE, the negative log likelihood was obtained by using the Bayesian GMM which was an outstanding technique for the multimodal distribution of the data with high computational efficiency [35,36].Among others, the PCA have been used successfully for object classification and scene classification.The PCA method is a statistical linear transformation selection from multiple variables to minor ones [28].Meanwhile, the GMM is a Gauss probability density model, which can be used to accurately quantify matters and classify them into several models based on the Gauss probability density function [23].
Taking advantages of the PCA and GMM methods in the processing of the multidimensional models, especially the reduction of the AE parameters and pattern recognitions, this paper intends to reduce the correlation dimension of the 15 parameters of the AE signals emitted from the damage process of the materials and to obtain the two new parameters which could be used to describe the overall damage property without linear dependence.Afterwards, the GMM was used to classify the damage into two major categories.Finally, the results were verified experimentally by using scanning electron microscopy image based classification.

Analysis to Construct the New Characterization
The PCA method shows obvious advantages in the multiparameter dimension reduction problems and the construction process is clear to operate.The observation matrix of the sample is discussed by Shang et al. [37], where the rows of the sample matrix  represent the AE parameters and the columns correspond to different signals.
The covariance matrix of the sample is which is the estimation of .Through the calculation of , the characteristic quantity of the original observation matrix can be easily reconstructed so as to facilitate the sort of the features.
Step 1.The covariance matrix of the sample is constructed by  as follows: where the matrix is a  ×  and positive definite matrix, and there are characteristic values of  which are not equal to each other and greater than zero.Each characteristic value corresponds to a unit feature vector.
Step 2. Compute the  features and its characteristic vector.Set  1 ,  2 , . . .,   to be eigenvalues of .Meanwhile,  =  1 ,  2 , . . .,   are the corresponding unit feature vectors.Arranging eigenvalues in a descending order gives Complexity 3 Step 3. Define the contribution rate of the characteristic value and the accumulated contribution rate Step 4. Based on the principle of the accumulated contribution ∑  =1   / ∑  =1   ≥ 85%, the former  ( ≤ ) principal components are picked, which means the former  mutually orthogonal eigenvector matrices are retained.
Step 5. Conduct linear correlation transformation between the new feature vector matrix and the original one.In this way, the original  dimension index will be reduced to , which contains the ultimate information with mutual linear independence.

Classification with GMM
The GMM probability density function is set as follows: where  is the mixed number of the model;   is the weighting coefficient of the model, and ∑   = 1; ( |   , ∑  ) is the th single Gauss probability density function, which is depicted as The proper parameters were evaluated as which makes the max maximum likelihood estimator of the probability density function, In order to obtain the maximum likelihood estimate, the GMM will be evaluated by the maximum expected value algorithm.The iteration steps are as follows.
Step 1. Initiate the parameters: (1) Set the mean values to be random values.
(3) Set the weighting coefficient  1 ,  2 ,  3 , . . .,   of each model to be the prior probability of each model: where  was the number of GMM.
Step 2. Compute the prior probability of each item in the model: Step 3. Update the parameters by the prior probability: Step 4. Repeat Steps 2 and 3 until the convergence: where  +1 and   are the parameters estimation of the previous and current step and  is the set threshold, which is usually set to 10 −5 .

Experimentation and Verification
4.1.Specimens and Experiment Setup.The industrial refractory tested in the study is composed of magnesia aggregates, carbon binder (phenolic resin and/or pitch), and other components.Figure 1 shows the microstructure of such a refractory without damage.The magnesia aggregates are formed by sintering of crystallites with weak interfaces.The size of magnesia grains varies from less than one half millimeters to five millimeters.The other grains impurities, such as SiO2 and Al2O3, with less than 5 mm sizes, are founded scattered in the matrix, the carbon binder.The components were mixed and shaped into bricks at low temperature (20∼50 ∘ C) and under high pressure (150 MPa around) [12].Then the bricks undergo heat treatment (100∼200 ∘ C) to start the polymerization of resin and to eliminate residual water and phenols [12].Under these sintering conditions, we directly obtain three specimens with a rectangular cross-section of 140 mm × 25 mm × 25 mm.Table 1 provides the composition of the materials after the heat treatment.According to ASTM C1161-13 [38], the shape of the samples for 3-point bending should be rectangular and the size of the samples should be 3 by 4 by 45 to 50 mm minimum with 40 mm outer span 3-point bending.Therefore, the specimens were prepared to be 140 mm × 25 mm × 25 mm.The 3-point bending tests were performed using a HMOR/STRAIN loading machine.The crosshead speed of the machine was fixed at 0.05 mm/min.The tests were executed on three specimens of each configuration in order to ensure the accuracy of the results (Figure 2).
AE is defined as phenomena whereby transient elastic waves are generated by the rapid release of energy from localized sources within a material (or structure).The AE, which represents the generation of transient ultrasonic waves due to damage development within the material under load, is an efficient technique for structural health monitoring, as discussed elsewhere [39][40][41].When a material is subjected to solicitations (such as mechanical and thermal), acoustic emission can be generated by a variety of sources, including crack nucleation and propagation, multiple dislocation slip, twinning, grain boundary sliding, phase transformations in alloys, debonding of grain in composite materials, or fracture of inclusions in alloys.This technique has been used at either the laboratory level or industrial scale.Usually, this technique is applied at room temperature as a nondestructive characterization technique in order to follow in real time the evolution of the damage of a material subjected to mechanical loading.Here, the upper surface of the sample should be slightly polished to remove the burr in order to locate the AE sensor.Then the coupling agent was coated on the polished zone and the AE sensor was fixed on the coupling agent surface with the adhesive tape.The stress wave was passed from the surface of the sample to the AE sensor through the coupling agent.The application of AE technique aims to characterize the material microdamage at a very local scale.
The device of acquisition (Figure 3) is composed of a wide band (175 kHz∼1 MHz) sensor (PAC MICROPHONE80), a preamplifier (EPA 1220A), and an acquisition card associated with a computer (AEDSP-32/16 MISTRAS digital system from Physical Acoustics Corporation).The AE sensor is a major element of the chain of acquisition because it collects the whole of the signals induced by the elastic waves created within the material whose amplitudes are higher than a fixed threshold in order to amplify and to record them.This system records the waveform and the main feature parameters well known in AE study such as count, hit, rise time, duration of hit, count to peak, and amplitude (in dB). Figure 4 presents different AE features extracted from the signal waveform.( 9 ), inverse calculation frequency ( 10 ), original frequency ( 11 ), signal strength ( 12 ), absolute energy ( 13 ), centroid frequency ( 14 ), and peak frequency ( 15 ).The number of the sample signals was 11168 and the observation matrix was 11168 × 15.In order to eliminate the disturbance of dimensionless parameter, the observation sample matrix was normalized before the principal component analysis and the data values were normalized to (0, 1).The covariance matrix's eigenvalues are shown in Table 2.The cumulative contribution rate of each principal component was shown in Table 3.It can be seen from the table that the cumulative contribution rate of the first two principal components is 90%, which is far greater than 85%, meaning the first two principal components are sufficient enough to replace the overall clustering index.Therefore, the new principal components are produced and the number of the parameters is reduced from 15 to 2.

Classification of the Damage Signals.
For the application of GMM classification of the damage signals of the refractory, the increase in the number of the model can improve the accuracy of the model, however with increased complexity of the model, as discussed by Jiang et al. [24].The Bayesian information criterion (BIC) has the ability to maintain the balance between the accuracy and complexity of the model; therefore, it is adopted to classify the damage.
where  is the maximum of the likelihood function of the estimated model,  is the number of observations, and  is the number of the free parameters to be estimated in each GMM.
When the number of the model is increased from  to  + 1, the changing rate of the BIC is The changing rate of the BIC reflects the sensitivity of the BIC values to the number of the models.When the number of the models is increased from  to  + 1, the change rate of the BIC is large, which means that the number  is insufficient in the description of the original data set accuracy and should be increased to +1.When the change rate of the BIC  is small,  and  + 1 have little difference in the description of the original data and  is enough for the description.The changing rate of the BIC is shown in Figure 5.It can be seen from the graph that when the number of the model is increased from 1 to 2, the change of the BIC is significant, reaching 7%.With the increase of the number of the models, the changing rate of the BIC gradually decreases (<3%).Therefore, the model number of 2 is chosen to describe the observed data set.
The GMM operation results are shown in Figure 6.It can be seen from the plot that the damage signal is divided into two categories of  1 and  2 , whose weights are 0.63 and 0.37, respectively.

Verification.
The Philips scanning electron microscopy (SEM, PSEM 500) and energy spectrometer (AMETEK) were The number of the model used for scanning analysis of the damage sample.The microscopic scanning results reveal that the damage forms were mainly the matrix and the interphase damage, as shown in Figures 7 and 8, and the matrix damage accounted for the larger proportion.The energy spectra results of the two kinds of microdamage are shown in Figures 9 and 10, respectively.It can be seen from Figure 9 that the main component of the matrix damage area is C with the mass fraction of 97% and the crack can be regarded as the matrix crack.From Figure 10, the composition of the observation area near the interface was C with the mass fraction of 65% and O and Mg with the mass fraction of 15% and 18%, respectively, which indicates that the matrix and the particle phase existed in the area and the crack is the interfacial crack.The SEM results show that the main damage form of the MgO-C refractory is the matrix and the interface damage and the matrix phase damage accounted for the larger proportion.Therefore, the classification results using the method of the PCA and GMM are verified.

Conclusions
In this paper, a new AE data processing method of merging the similar parameters in the description of damage to reduce the dimension was developed.In the proposed method, the AE damage signals of the Mg-O refractory during the three-point bend test were collected and 15 parameters were directly obtained: rise time ( 1 ), count ( 2 ), energy ( 3 ), duration ( 4 ), amplitude ( 5 ), mean frequency ( 6 ), RMS ( 7 ), ASL ( 8 ), peak frequency ( 9 ), inverse calculation  ).The observation sample matrix was firstly normalized before the principal component analysis and the data values were normalized to (0, 1).The cumulative contribution rate of each principal component was calculated to successfully select the first two principal components of 90% contribution.Therefore, the new principal components were produced and the number of the parameters was reduced from 15 to 2.
Then the Gaussian mixture model was used to classify the damage of the refractory according to the 2 damage indexes, which could be utilized to describe the overall damage property without linear dependence.Afterwards, the damage was classified into two major categories of  1 and  2 with the damage weight of 63% and 37%, respectively.In order to verify the proposed method, the Philips scanning electron microscopy and energy spectrometer were used for scanning analysis of the sample.The scanning results showed that the damage form was indeed observed as 2 damage forms of mainly the matrix and the interphase damage.The main component of the matrix damage area was C with the mass fraction of 97%.In the interface damage crack area, C was with the mass fraction of 65%, and O and Mg were with the mass fraction of 15% and 18%, respectively, which indicated that the matrix and the particle phase existed in the area and the crack was the interfacial crack.At last, the SEM results showed that the main damage form of the MgO-C refractory was the matrix and the interface damage and the matrix phase damage accounted for the larger proportion.Therefore, the classification results using the method of the PCA and GMM were verified.

Figure 2 :Figure 3 :
Figure 2: Experimental setup for three-point bend test at room temperature.(a) The physical diagram and (b) the schematic one.The AE sensor is fixed on the specimen through the adhesive tape.The pressure actuating pressure head is imposed on the middle of the specimen.

Figure 5 :Figure 6 :
Figure 5: BIC change rate of each model number.

Figure 10 :
Figure 10: EDS analysis of the matrix damage crack.

Table 1 :
The composition of magnesia carbon refractory.
Figure 1: Typical microstructure of magnesia carbon refractory.

Table 2 :
Eigenvalues of covariance matrix.

Table 3 :
Accumulated contribution rate of each component.
Figure 9: EDS analysis of the matrix damage crack.